Noname manuscript No. (will be inserted by the editor)

Higher torsion in p-groups, Casimir operators and the classifying spectral sequence of a Lie algebra

Jonathan Pakianathan · Nicholas Rogers

Received: date / Accepted: date

Abstract We study exceptional torsion in the integral cohomology of a family of p-groups associated to p-adic Lie algebras. A spectral sequence E_{r}^{∗,∗}[𝔤] is defined for any Lie algebra 𝔤 which models the Bockstein spectral sequence of the corresponding group in characteristic p. This spectral sequence is then studied for complex semisimple Lie algebras like 𝔰𝔩_{n}(ℂ), and the results there are transferred to the corresponding p-group via the intermediary arithmetic Lie algebra defined over ℤ.

Over ℂ, it is shown that E_{1}^{∗,∗}[𝔤] = H^{∗}(𝔤, U(𝔤)^{∗}) = H^{∗}(ΛBG) where U(𝔤)^{∗} is the dual of the universal enveloping algebra of 𝔤 and ΛBG is the free loop space of the classifying space of a Lie group G associated to 𝔤. In characteristic p, a phase transition is observed. For example, it is shown that the algebra E_{1}^{∗,∗}[𝔰𝔩_{2}[𝔽_{p}]] requires at least 17 generators unlike its characteristic zero counterpart which only requires two.

Keywords Lie algebra · cohomology · p-group · free loop space · Bockstein spectral sequence

Mathematics Subject Classification (2000) 20J06 · 17B56 · 55T05 Contents

J. Pakianathan Dept. of Mathematics University of Rochester Rochester, NY 14627 U.S.A.

Tel.: +1-585-275-2216 Fax: +1-585-273-4655 E-mail: jonpak@math.rochester.edu N. Rogers Dept. of Mathematics University of Rochester Rochester, NY 14627 U.S.A.

Tel.: +1-585-275-9418 E-mail: rogers@math.rochester.edu

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 The Classifying Spectral Sequence E[𝔤] and Universal Coefficient Arguments … 9 3 Computation of Dual Casimirs and Invariant Theory . . . . . . . . . . . . . . . . . 13 4 Computation of E_{r}^{∗,∗}[𝔤] where 𝔤 is a complex simple Lie algebra . . . . . . . . . . 17 5 Exponent theory for finite p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6 Non-abelian Lie algebra of dimension 2 . . . . . . . . . . . . . . . . . . . . . . . . . 28 7 Weight Stratification of the Spectral Sequence E_{r}^{∗,∗} . . . . . . . . . . . . . . . . . . 31 8 Witt Lie algebras and Algebraic de Rham Cohomology . . . . . . . . . . . . . . . . 33 9 Computation of E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 10 Bounds on the exceptional torsion in congruence subgroups . . . . . . . . . . . . . 48 11 Comments on Modular Forms and Orthogonal Steenrod Structures . . . . . . . . . 48 12 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A Construction of the Algebraic Spectral Sequence E[𝔤] . . . . . . . . . . . . . . . . 50

1 Introduction

Let k be a PID throughout this paper (in a lot of cases we’ll be using a field but sometimes we need k = ℤ or other decent rings and still use the term algebra through abuse of notation). By a k-Lie algebra 𝔤 we mean a free kmodule equipped with a bilinear bracket [−, −] : 𝔤 ⊗_{k} 𝔤 → 𝔤 satisfying the Jacobi identity:

[[x, y], z] + [[y, z], x] + [[z, x], y] = 0

for all x, y, z ∈ 𝔤. We refer to the dimension of 𝔤 as the rank of the underlying free k-module, and will usually assume this is finite unless otherwise specified.

Lie algebras arise in a variety of contexts: over ℝ as the tangent space at the identity of a Lie group or as the collection of smooth vector fields on a smooth manifold, over ℚ as the rational homotopy Lie algebra of a space, over ℤ as the Lie algebra associated to the descending central series of a residually nilpotent group and over the p-adic integers ℤ_{p} or finite fields in the theory of pro-p groups (see [DS]) and p-groups respectively. For basic facts on Lie algebras and their cohomology used in this paper see [Bo] or [GG]. For the basic facts on the cohomology of groups used in this paper see [Be] or [B].

Due to our primary motivation let us give a few more details on the correspondence between p-groups and Lie algebras. Consider the formal power series for e^{x} in ℚ[[x]], i.e., e^{x} = ∑_{n=0}^{∞} \frac{x^{n}}{n!} , it is a trivial formality to check e^{0} = 1 and e^{−x}e^{x} = 1 and that if x and y commute then e^{x+y} = e^{x}e^{y}. However working in the (completed) free associative algebra over ℚ on two variables one finds that if x and y don’t commute, then one has the fundamental Baker-Campbell-Hausdorff identity:

e^{x}e^{y} = e^{x+y+ \frac{1}{2} [x,y]+ \frac{1}{12} [x,[x,y]]− \frac{1}{12} [y,[x,y]]+...} = e^{x+y+I}

where I is an infinite sum of iterated brackets of x and y with rational coefficients and where [a, b] = ab − ba. Due to this, whenever ℚ ⊆ k, we can associate a formal group e^{𝔤} to every k-Lie algebra 𝔤. When k = ℂ and the usual Euclidean metric is used, the defining series converge and one recovers the classical “Exponential-Log” correspondence between Lie algebras and Lie groups.

Over the p-adic integers ℤ_{p} with the p-adic metric, and p an odd prime, e^{px} can be shown to converge for every x ∈ ℤ_{p}. Thus if we let 𝔤 = 𝔤𝔩_{n}(ℤ_{p}) be the Lie algebra of n × n, p-adic matrices with the bracket [𝔸, 𝔹] = 𝔸𝔹 − 𝔹𝔸, one has that e^{p𝔤} is a group. In fact since e^{p𝔸} ≡ 𝕀 mod p, it is not hard to show that e^{p𝔤} is the group of p-adic matrices which are congruent to the identity matrix mod p, which is called the p-congruence subgroup Γ_{𝔤𝔩_{n}} of GL_{n}(ℤ_{p}). (See exponential-log correspondence in [Ro]). From this formal group, one can form a tower of finite p-groups Γ_{𝔤,k} = e^{p𝔤} mod p^{k+1} for k = 1, 2, 3, … for which e^{p𝔤} is the inverse limit. It is not hard to see that Γ_{𝔤,1} is elementary abelian and can be identified naturally with the residue Lie algebra of the p-adic Lie algebra 𝔤 i.e., 𝔤/p𝔤 = 𝔤 ⊗ 𝔽_{p}. In fact e^{p𝔸} = 𝕀 + p𝔸 mod p^{2} so e^{p𝔸} mod p^{2} corresponds to 𝔸 mod p. A little more analysis shows that we have a central short exact sequence

0 → 𝔤 ⊗ 𝔽_{p} → Γ_{𝔤,2} → 𝔤 ⊗ 𝔽_{p} → 0.

and so Γ_{𝔤,2} = Γ_{𝔤} mod p^{3} is a p-group given by a central extension of an elementary abelian p-group by itself.

More generally, for p odd, one can show given any 𝔽_{p}-Lie algebra 𝔏 (whether it lifts to the p-adics or not) there exists a unique p-power exact sequence

0 → 𝔏 → G(𝔏) → 𝔏 → 0

where G(𝔏) is a p-group of order p^{2 dim(𝔏)}. In fact the construction 𝔏 → G(𝔏) is part of a covariant functor from the category of 𝔽_{p}-Lie algebras to the category of p-groups. (See [BP]). For p = 2 certain phenomena involving quadratic forms arise and the correspondence needs to be modified (see [PY1] and [PY2]). Due to this, throughout this paper, p denotes an odd prime.

In [BP] it was shown that if n is the dimension of 𝔏, then

H^{∗}(G(𝔏), 𝔽_{p}) ≅ Λ^{∗}(𝔏^{∗}) ⊗ P oly(𝔏^{∗}) ≅ Λ(x_{1}, …, x_{n}) ⊗ 𝔽_{p}[y_{1}, …, y_{n}]

where 𝔏^{∗} is the dual of 𝔏. Here Λ^{∗}(V) denotes the exterior algebra on the vector space V where V is given grading 1 while P oly(V) = 𝔽_{p}[V] denotes the polynomial algebra on V where V is given grading 2.

Furthermore the differential provided by the Bockstein β was shown to be the same as the differential that makes this algebra into the Koszul resolution computing H^{∗}(𝔏, P oly(ad^{∗})) where ad^{∗} is the dual adjoint representation of 𝔏 on 𝔏^{∗} and the action of 𝔏 is extended to P oly(ad^{∗}) by declaring it to act via derivations.

Thus the β-cohomology of these p-groups was shown to be H^{∗}(𝔏, P oly(ad^{∗})) and this is interesting as this cohomology gives the 2nd page of the Bockstein spectral sequence used to analyze the integral cohomology H^{∗}(G(𝔏); ℤ). Explicitly if we decompose the polynomial algebra P oly(ad^{∗}) into its homogeneous components (Hodge decomposition) P oly(ad^{∗}) = ⊕_{n=0}^{∞}S^{n} we have

B_{2}^{∗} = ⨁_{n=0}^{∞} H^{∗}(𝔏, S^{n})

and so the higher torsion in the integral cohomology of the p-group G(𝔏) is reflected in these Lie algebra cohomology groups.

In fact an analysis of the integral cohomology of G(𝔰𝔩_{2}(𝔽_{p})) using these techniques was used to provide a counterexample in [Pk] to a conjecture of Adem (see [A]) at odd primes.

In this paper, we study a fundamental differential graded algebra (dga) associated to any Lie algebra 𝔏 given by the Koszul resolution whose underlying algebra is Λ^{∗}(𝔏^{∗}) ⊗ P oly(𝔏^{∗}) with a differential such that the cohomology calculates H^{∗}(𝔏, P oly(ad^{∗})). While our motivation is to calculate higher torsion in the integral cohomology of p-groups and hence primarily concerns Lie algebras over 𝔽_{p}, our basic technique is to relate these Lie algebras to Lie algebras defined over ℤ and to then compare these to the corresponding Lie algebras defined over ℚ or ℂ where classical results can be appealed to. This translates then to results which hold for “all but finitely many primes” when studying the corresponding p-groups. This philosophy was motivated by previous work with A. Adem in [AP], though the implementation is quite different here.

The dga we construct E[𝔏] = Λ^{∗}(𝔏^{∗}) ⊗ P oly(𝔏^{∗}) comes equipped with a spectral sequence that is constructed in the appendix and functions in some sense as an algebraic classifying complex for the co-Lie algebra 𝔏^{∗} much like EG does for a Lie group G.

Interestingly enough, this seemingly specialized dga carries a lot of structure. Over fields of characteristic zero and for nondegenerate Lie algebras (Killing form nondegenerate), one has an isomorphism of 𝔏-modules between ad and ad^{∗}. Furthermore using the Poincare-Birkoff-Witt theorem one can identify P oly(ad) with U(𝔏) as filtered modules and P oly(ad^{∗}) with U(𝔏)^{∗} as graded modules where U(𝔏) is the universal enveloping algebra of 𝔏 equipped with the adjoint action. (Throughout this paper U(𝔏) and its dual will always be equipped with the adjoint action and not the left or right translation action!) Thus E[𝔏] as a dga is nothing other than the canonical Koszul complex computing H^{∗}(𝔏, U(𝔏)^{∗}). In this context it is important to point out that over fields of characteristic zero, H^{∗}(𝔏, U(𝔏)^{∗}) can be identified with the cohomology of the free loop space of BG in the case that G is a compact, connected Lie group with Lie algebra a real form for 𝔏 as we will see in this paper in the case that G is semisimple as a byproduct of our analysis of this spectral sequence.

The cohomology of the free loop space ΛM on a manifold M has been an object of intense study in the last decade due to the existence of a “string multiplication” introduced by Chas and Sullivan (See [CS]) and the structure of a Batalin-Vilkovisky algebra with string theoretic interpretations, (see [GW], [Ma]).

Furthermore H^{0}(𝔏, U(𝔏)) consists of the central elements of U(𝔏), the so called “Casimir ring” which is very important in the representation theory of 𝔏. We will see that their dual elements, the “dual Casimirs” in H^{0}(𝔏, U(𝔏)^{∗}) play an important role in the higher torsion of p-groups.

As one of the main tools of this paper, we construct a spectral sequence whose E_{0} term is this dga and whose E_{∞} term is the cohomology of a point which gives a lot of structure to the underlying dga. Though this spectral sequence is defined over any coefficient k, over 𝔽_{p} it models the associated Bockstein Spectral sequence of the corresponding p-group mentioned above. More precisely we show that the (E_{0}, d_{0}) term of this spectral sequence is identical to the (B_{1}, β) term of the Bockstein spectral sequence of the p-group G(𝔏). Though both spectral sequences converge to the cohomology of a point, it is unknown if the higher pages coincide. Nevertheless, using the structure theorems available for both spectral sequences, and that the first one can be used to compare the situation for the 𝔽_{p}-Lie algebra with that of corresponding integral and complex Lie algebras, one can obtain results. Throughout this paper (E_{r}, d_{r}) will always refer to the algebraic spectral sequence constructed in the appendix and (B_{r}, β_{r}) will denote the Bockstein spectral sequence of the group G(𝔏) in the case that 𝔏 is a 𝔽_{p}-Lie algebra.

The spectral sequence E_{r} carries a lot of information, indeed even over ℂ it can be used to recover Harish-Chandra’s calculation of the Casimir ring of sl_{2}(ℂ) among many other results. (We discuss the physical relevance of this within the paper.) In this introduction we quote just this classical corollary for simplicity to show the main frame of the arguments. In some of these statements the “Hodge decomposition” P oly(ad^{∗}) = ⊕_{n=0}^{∞}S^{n} of the polynomial algebra into its homogeneous components is used and S^{0} always denotes the base ring k with trivial 𝔏 action.

Theorem 1 (Harish-Chandra calculation for 𝔰𝔩_{2}(ℂ))

H^{∗}(𝔰𝔩_{2}(ℂ), P oly(ad^{∗})) ≅ H^{∗}(𝔰𝔩_{2}(ℂ), U(𝔰𝔩_{2}(ℂ))^{∗}) ≅ Λ^{∗}(u) ⊗ ℂ[κ]

where κ = H^{2} + EF ∈ H^{0}(𝔰𝔩_{2}(ℂ), S^{2}) is (a nonzero scalar multiple) of the Killing form (an example of a “dual Casimir” element) and u ∈ H^{3}(𝔰𝔩_{2}(ℂ), S^{0}) is the volume form. κ is dual to the central Casimir element H^{2} + 2EF + 2F E in H^{0}(𝔰𝔩_{2}, U(𝔰𝔩_{2})) which corresponds to “total angular momentum squared” in spin systems.

As mentioned before, this calculation when used in conjunction with the spectral sequence we construct can be used to derive results for the corresponding Lie algebras 𝔰𝔩_{2}(𝔽_{p}) after passing through 𝔰𝔩_{2}(ℤ). To state these results let us note that for 𝔽_{p}-Lie algebras 𝔏 we have (E_{0}, d_{0}) = (B_{1}, β) = Λ^{∗}(𝔏^{∗}) ⊗ P oly(𝔏^{∗}) where (B_{r}, β_{r}) is the Bockstein spectral sequence of the p-group G(𝔏). We will call the polynomial degree of a homogeneous element of this algebra its “Hodge degree”.

This leads to the following:

Theorem 2 (Higher torsion in G(𝔰𝔩_{2}(𝔽_{p}))) Let G(𝔰𝔩_{2}(𝔽_{p})) be the kernel of the reduction homomorphism SL_{2}(ℤ/p^{3}ℤ) → SL_{2}(𝔽_{p}). Then if B^{∗} denotes the Bockstein spectral sequence for G(𝔰𝔩_{2}(𝔽_{p})) we have:

E_{1}^{∗,∗} = B_{2}^{∗} = H^{∗}(𝔰𝔩_{2}(𝔽_{p}), P oly(ad^{∗})) ∼ Λ^{∗}(u) ⊗ 𝔽_{p}[κ]

where ∼ denotes isomorphism in the range of Hodge degree ≤ N for all but finitely many primes p (depending on N).

Furthermore B_{3}^{∗} is finite dimensional and we have B_{3}^{∗} ∼ Λ^{∗}(u) ⊗ 𝔽_{p}[κ]. In particular though B_{3}^{∗} is always finite dimensional, there is no fixed bound on its dimension that holds for all primes p.

Let us try to explain this in a less technical manner. For a finite p-group G let exp(G) be the exponent of G, i.e., the smallest positive integer n such that g^{n} = e for all g ∈ G. Let H̄^{∗}(G, ℤ) denote the reduced integral cohomology of G and e(G) denote its exponent. Let e_{∞}(G) denote the asymptotic exponent of G, i.e., the smallest positive integer such that e_{∞}(G)H̄^{∗}(G, ℤ) is finite. It is known that

exp(G) | e_{∞}(G) | e(G) | |G|

and there exist G such that e_{∞}(G) ≠ e(G). (See [Pk]). When e_{∞}(G) ≠ e(G), we can define the highest dimension of an element in the finite graded group e_{∞}(G)H̄^{∗}(G, ℤ) as the “exceptional dimension” of G. Thus all torsion elements of exceptionally high order lie at or below the exceptional dimension of G.

Corollary 1 For all odd primes p, e_{∞}(G(𝔰𝔩_{2}(𝔽_{p}))) = p^{2} while e(G(𝔰𝔩_{2}(𝔽_{p}))) = p^{3}. Moreover by the calculations above, for any N, for all but a finite number of primes p, the exceptional dimension of G(𝔰𝔩_{2}(𝔽_{p})) is bigger than N.

Thus for every odd prime there are exceptional elements of order p^{3} in the reduced integral cohomology of G(𝔰𝔩_{2}(𝔽_{p})), but there is no bound on the exceptional dimension that holds for all primes. Thus by suitable choice of primes p, one can find elements of order p^{3} in as high a dimension as one likes. However, for any fixed prime p, asymptotically the exponent of the integral cohomology of G(𝔰𝔩_{2}(𝔽_{p})) is always p^{2}.

For details on these calculations and their implications a more thorough and leisurely development can be found in the paper itself. The basic idea is as follows:

(1) For every simple Lie algebra over ℂ, a Cartan-Serre basis can be taken to extract a corresponding integral Lie algebra. The algebraic spectral sequences for these can be compared and, through characteristic zero techniques, computations of the required dual Casimirs can be done.

(2) Since each piece in the Hodge decomposition of the dga of the corresponding integral Lie algebra is of finite type, when looking at a chunk corresponding to terms with Hodge degree N or less, one can use the universal coefficient theorems to say that for all but a finite number of primes, the corresponding dga over 𝔽_{p} will look similar to the one over ℂ.

(3) In every case though there is a breakdown in the dga over 𝔽_{p} when the Hodge degree becomes close to p − 1 and there is generally a phase transition between characteristic zero behaviour and characteristic p behaviour.

(4) In many cases we find that the “exceptional torsion” is created by the part which corresponds to the characteristic zero case and the transition to characteristic p kills this exceptional torsion and is signaled by a divided power issue in the dga for the integral Lie algebra. For example for 𝔰𝔩_{2} we prove the key identity κ^{\frac{p−1}{2}} u = 0 in the 𝔽_{p}-dga as the left hand side of the identity is p times a generator in the corresponding ℤ-dga. Detailed pictures of the algebraic SS are available in the paper to show how this identity helps cause a “phase-transition” between the char 0 and char p behaviour in the case of 𝔰𝔩_{2}.

In the paper, all semisimple Lie algebras are studied, including 𝔰𝔩_{n} for n > 2, and the exceptional Lie algebra 𝔤_{2}. These are studied over ℂ, ℤ and finite fields. Over ℂ one finds the behaviour of E_{r}^{∗,∗}[𝔤] is like that of EG which motivates us calling it the “classifying spectral sequence for the Lie algebra” in general.

Theorem 3 (Complex semisimple Lie algebras) For any complex simple Lie algebra 𝔤 with corresponding compact form 𝔤_{ℝ} and compact connected Lie group G with Lie algebra 𝔤_{ℝ} we have:

E_{1}^{∗,∗}[𝔤] = H^{∗}(𝔤, U(𝔤)^{∗}) ≅ H^{∗}(G, ℂ) ⊗ H^{∗}(BG, ℂ) ≅ H^{∗}(ΛBG, ℂ).

where ΛBG denotes the free loop space of the classifying space BG. For example, for 𝔤 = 𝔰𝔩_{n}(ℂ) one has

E_{1}^{∗,∗} = H^{∗}(𝔰𝔩_{n}, U(𝔰𝔩_{n})^{∗}) ≅ H^{∗}(SU(n), ℂ) ⊗ H^{∗}(BSU(n), ℂ)
≅ H^{∗}(ΛBSU(n), ℂ)
≅ Λ^{∗}(u_{3}, …, u_{2n−1}) ⊗ ℂ[c_{2}, …, c_{n}]
≅ Λ^{∗}(u_{3}, …, u_{2n−1}) ⊗ ℂ[σ_{2}, …, σ_{n}]

where c_{i} are the universal Chern classes and σ_{i} are the adjoint invariant polynomial functions on 𝔰𝔩_{n} given by the elementary symmetric functions on the eigenvalues expressed in terms of the coefficients of the matrix. This invariant theory picture is used to obtain these results and is explained completely in the relevant sections of the paper. In general, there are 3 core pictures for E_{1}^{∗,0} = H^{0}(𝔤, U(𝔤)^{∗}) discussed in this picture, (a) as elements dual to central elements in U(𝔤), i.e., as dual Casimirs, (b) as adjoint invariant polynomial functions on 𝔤 and (c) as the cohomology algebra H^{∗}(BG, ℂ).

As a final example, for 𝔤 = 𝔰𝔭_{2n}(ℂ) one has

\begin{aligned}E_{1}^{∗,∗} = H^{∗}(𝔰𝔭_{2n}(ℂ), U(𝔰𝔭_{2n}(ℂ))^{∗}) & ≅ H^{∗}(ΛBSp(n), ℂ) \\ & ≅ Λ^{∗}(u_{3}, u_{7}, …, u_{4n−1}) ⊗ H^{∗}(P_{1}, P_{2}, …, P_{n}) \end{aligned}

where P_{i} are the universal Pontryagin classes.

In general, (E_{0}^{∗,∗}[𝔤], d_{0}) is a free graded commutative dga whose cohomology is that of the free loop space of BG. (However it is not a Sullivan algebra as it fails to satisfy the nilpotency condition.)

As mentioned above, each of these results over ℂ yield a picture for the spectral sequence of the associated Lie algebra over 𝔽_{p} and hence the Bockstein spectral sequence of the corresponding p-group, at least for low Hodge degree and for all but finitely many primes. This is discussed in detail in the paper, as is the behaviour of higher pages of the spectral sequence.

Over ℂ, these results are probably, by and large, repackaging of classical results in the cohomology of Lie groups and their classifying spaces, invariant theory and Casimir theory into a spectral sequence, but we go through the process in detail in the paper as we need the spectral sequence for our work in p-groups.

When working in prime characteristic, computations become much more difficult and a “weight stratification” is required. If R is a system of roots for the semisimple Lie algebra 𝔤 with root lattice Λ(R), we show that there is a decomposition of the spectral sequence by weight which is very helpful for computations of associated Lie algebras in prime characteristic:

Theorem 4 (Weight Stratification) Let 𝔤 be a complex semisimple Lie algebra. Let 𝔤_{ℤ} be a corresponding integral Lie algebra (always exists by Cartan-Serre basis). We can then get a corresponding Lie algebra 𝔤_{k} over any ring of definition k.

The root decomposition of 𝔤 induces a weight decomposition of spectral sequences:

E_{r}^{∗,∗}[𝔤_{k}] = ⨁_{α∈Λ(R)} E_{r}^{∗,∗}[α]

where Λ(R) is the root lattice of 𝔤 and is a free abelian group of rank equal to the rank of 𝔤, i.e., the dimension of a Cartan subalgebra of 𝔤.

In addition we show that if k is a field of characteristic zero then the E_{1}-page and beyond only has contributions from the weight 0 term while for a field of characteristic p, the “polynomial line” E_{1}^{∗,0} has contributions only from weights which are zero modulo p.

In sections 8 and 9, using a DeRham complex and D-module language together with the weight stratification mentioned above, explicit mod p calculations are performed for the spectral sequence E_{r}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] and the “almost all” caveat is removed in a range of Hodge degrees:

Theorem 5 (𝔰𝔩_{2}(𝔽_{p})-computation)

E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] = H^{∗}(𝔰𝔩_{2}(𝔽_{p}), P oly(ad^{∗})) ∼ Λ^{∗}(u) ⊗ 𝔽_{p}[κ]

where ∼ indicates isomorphism for all odd primes in the range of Hodge degrees strictly less than p − 1. Thus for all odd primes p, the characteristic 0 to p phase transition does not occur before Hodge degree p − 1 for the Lie algebra scheme 𝔰𝔩_{2}(−). (For p = 2 the breakdown occurs immediately at Hodge degree 0.)

However a char 0 to char p phase transition occurs at Hodge degree p − 1 and it is shown that one needs minimally 17 generators to generate H^{∗}(𝔰𝔩_{2}(𝔽_{p}), P oly(ad^{∗})) through Hodge degree p. (see section 9 for details on these generators and the algebra).

It follows trivally that H^{∗}(𝔰𝔩_{2}(ℤ), P oly(ad^{∗})) has p-torsion for every prime p and hence cannot be a finitely generated ring.

From this it follows that the exceptional dimension (maximal dimension for which exceptional high order torsion exists) for the p groups G(𝔰𝔩_{2}(𝔽_{p})) is greater than or equal to 2p − 2 for all odd primes p.

As a byproduct of the analysis of the spectral sequence, the following exact sequence is obtained for 𝔰𝔩_{2} = 𝔰𝔩_{2}(𝔽_{p}) and i ≥ p − 2:

0 → H^{3}(𝔰𝔩_{2}, S^{i}) → H^{2}(𝔰𝔩_{2}, S^{i+1}) → H^{1}(𝔰𝔩_{2}, S^{i+2}) → H^{0}(𝔰𝔩_{2}, S^{i+3}) → 0,

where S^{i} is the module of homogeneous, degree i polynomials on 𝔰𝔩_{2} equipped with the dual adjoint action.

Finally while many of the techniques apply to solvable (or nilpotent) Lie algebras also, we concentrate on semisimple Lie algebras in this paper, postponing the discussion for nilpotent Lie algebras such as those which would arise from torsion-free, pro-p, finite index, subgroups of the Morava Stabilizer group until another time.

2 The Classifying Spectral Sequence E[𝔤] and Universal Coefficient Arguments

Associated to any k-Lie algebra 𝔤 is a spectral sequence (E[𝔤]_{r}^{s,t}, d_{r}), constructed in the appendix. We call t the exterior degree and s the Hodge or polynomial degree and 2s + t the total degree. This is a first quadrant spectral sequence and often we will plot the Hodge degree along the x-axis and either the exterior degree t or the total degree 2s + t along the y-axis. The reader is encouraged to draw such a diagram when following the arguments. Often we will surpress 𝔤 explicitly from the notation when it is understood. This spectral sequence has the following properties:

(1) As a k-algebra E_{0}^{∗,∗} = Λ^{∗}(𝔤^{∗}) ⊗ k[𝔤^{∗}]; i.e., it is the tensor product of the exterior algebra on the dual of 𝔤 with the polynomial algebra on the dual of 𝔤. If s denotes the natural isomorphism Λ^{1}(𝔤^{∗}) → P oly^{1}[𝔤^{∗}] = 𝔤^{∗}, then we can write E_{0}^{∗,∗} = Λ^{∗}(𝔤^{∗}) ⊗ k[s(𝔤^{∗})] as a bigraded algebra where 𝔤^{∗} is given bigrading (s, t) = (0, 1) while s(𝔤^{∗}) is given bigrading (s, t) = (1, 0). Note however that graded commutativity of the dga is determined by total degree and that the elements s(𝔤) have even total degree as required for them to generate a polynomial algebra. Explicitly, if {e_{1}, …, e_{n}} is a k-basis for 𝔤 and {x_{1}, …, x_{n}} the canonical dual basis for 𝔤^{∗} determined by x_{i}(e_{j}) = δ_{i,j}, and we set y_{i} = s(x_{i}), then

E_{0}^{∗,∗} = Λ^{∗}(x_{1}, …, x_{n}) ⊗ k[y_{1}, …, y_{n}].

(2) Each page of the spectral sequence is a dga over k such that

d_{r} : E_{r}^{s,t} → E_{r}^{s+r,t−(2r−1)}

i.e., the rth differential raises the Hodge degree by r and reduces the exterior degree by (2r − 1), and hence raises total degree by 1. The differential d_{r} is a derivation with respect to the induced algebra structure; i.e.,

d_{r}(αβ) = d_{r}(α)β + (−1)^{|α|}αd_{r}(β),

where α, β are homogeneous elements of E_{r}^{∗,∗} and |α| denotes the total degree of α.

Of course as usual E_{r+1}^{∗,∗} = H^{∗}(E_{r}^{∗,∗}, d_{r}).

(3) The differential d_{0} is induced naturally as follows. It is the unique derivation on E_{0}^{∗,∗} such that

d_{0} : Λ^{1}(𝔤^{∗}) → Λ^{2}(𝔤^{∗})

is minus the dual of the Lie-bracket [−, −] : Λ^{2}(𝔤) → 𝔤 and

d_{0} : s(𝔤^{∗}) → s(𝔤^{∗}) ⊗ Λ^{1}(𝔤^{∗})

is the dual of the Lie-bracket followed by Identity ⊗ s^{−1}. More explicitly if c_{ij}^{k} are the structure constants of 𝔤 with respect to the k-basis {e_{1}, …, e_{n}}, {x_{i}} is the dual basis to {e_{i}}, and y_{i} = s(x_{i}), then

\begin{matrix}d_{0}(x_{i}) = − ∑_{j<k} c_{jk}^{i}x_{j}x_{k} and \\ d_{0}(y_{i}) = ∑_{1≤j,k≤n} c_{jk}^{i}y_{j}x_{k}. \end{matrix}  (1)

(4) Thus the dga (E_{0}^{∗,∗}, d_{0}) breaks up as a direct sum of finite dimensional dga’s E_{0}^{∗,∗} = ⊕_{s=0}^{∞}(E_{0}^{s,∗}, d_{0}) where the decomposition is based on polynomial degree and will be refered to as the “Hodge decomposition”. As explained in the appendix, each of these turns out to be the Koszul resolution for the Lie algebra 𝔤 for the module P oly^{s}(𝔤^{∗}) of homogeneous degree s polynomials on the dual of 𝔤. (For infinite fields, P oly^{s}(𝔤^{∗}) can be identified as homogeneous degree s polynomial functions f : 𝔤 → k on 𝔤.) Thus one has

E_{1}^{∗,∗} = ⨁_{s=0}^{∞} H^{∗}(𝔤, P oly^{s}(𝔤^{∗})) = H^{∗}(𝔤, P oly(𝔤^{∗})).

When k is a field of characteristic zero, one can identify the dual of the Universal Enveloping algebra of 𝔤 i.e., U(𝔤)^{∗} with P oly(𝔤^{∗}) as an algebra where U(𝔤)^{∗} is given the dual algebra structure coming from the comultiplication in the primitively generated Hopf algebra U(𝔤). This follows essentially from the Poincare-Birkoff-Witt theorem (we need a field of characteristic zero as in general the dual of a primitively generated Hopf algebra is a divided power algebra which only is a polynomial algebra over fields of characteristic zero). This isomorphism is also one of 𝔤-modules as long as we equip U(𝔤)^{∗} with the dual adjoint action coming from the adjoint action of 𝔤 on U(𝔤). Thus over fields of characteristic zero,

E_{1}^{∗,∗} = H^{∗}(𝔤, U(𝔤)^{∗}).

(5) The spectral sequence comes from two anticommuting differentials on E_{0}^{∗,∗}, d_{0} and d_{1}. d_{1} is defined as follows: d_{1} is the unique derivation which is s on Λ^{1}(𝔤^{∗}) and zero on P oly^{1}(s(𝔤^{∗})) i.e., d_{1}(x_{i}) = y_{i} and d_{1}(y_{i}) = 0 for all 1 ≤ i ≤ n. This d_{1} induces the differential on E_{1}^{∗,∗}. Then via a completely standard zig-zag construction (see [BT]), all higher differentials can be determined from these two as follows: If ω represents a class in E_{r} then one inductively can define d_{1}(ω) = d_{0}(α_{1}), d_{1}(α_{k}) = d_{0}(α_{k+1}) for 1 ≤ k ≤ r − 2 and finally d_{1}(α_{r−1}) represents d_{r}(ω) and, as is typical with these sorts of constructions, the final class can be shown to be independent of the choices made along the way.

(6) Once 2r − 1 > dim(𝔤) we have d_{r} = 0 and so E_{m}^{∗,∗} = E_{∞}^{∗,∗} once m > \frac{dim(𝔤)+1}{2}. Furthermore as shown in the appendix,

E_{∞}^{s,t} = \begin{cases}k if s = t = 0; \\ 0 otherwise. \end{cases}

(7) E[−]^{∗,∗} defines a contravariant functor from the category of k-Lie algebras to the category of k-differential graded algebra spectral sequences. In other words, if θ : 𝔤 → 𝔥 is a map of k-Lie algebras then it is not hard to show that θ^{∗} : 𝔥^{∗} → 𝔤^{∗} commutes with the dual Adjoint action in the sense that if we make 𝔥^{∗} a 𝔤-module using θ then θ^{∗} is a 𝔤-module map. It is then a routine exercise to check that the algebra map induced from θ,

Λ^{∗}(𝔥^{∗}) ⊗ k[s(𝔥^{∗})] → Λ^{∗}(𝔤^{∗}) ⊗ k[s(𝔤^{∗})],

commutes with d_{0} and d_{1} and hence induces a morphism of spectral sequences (E[𝔥]_{r}^{∗,∗}, d_{r}) → (E[𝔤]_{r}^{∗,∗}, d_{r}).

(8) The construction E[−]^{∗,∗} is natural with respect to extension of scalars. In other words if k → K is a homomorphism of rings between two PIDs and 𝔤 is a k-Lie algebra then 𝔤 ⊗_{k} K is naturally a K-Lie algebra and

(E[𝔤 ⊗_{k} K]_{0}^{∗,∗}, d_{0}) = (E[𝔤]_{0}^{∗,∗} ⊗_{k} K, d_{0} ⊗_{k} Id)

as K-dga’s. Thus if K is flat over k (this is needed to avoid tor terms in the universal coefficient theorem) one has

(E[𝔤 ⊗_{k} K]_{r}^{∗,∗}, d_{r}) = (E[𝔤]_{r}^{∗,∗} ⊗_{k} K, d_{r} ⊗_{k} Id)

as K-spectral sequences.

(9) For k a field with p elements, p an odd prime and 𝔤 a k-Lie algebra which lifts over ℤ/p^{2}ℤ, one has the following result proven in [BP]:

If (B_{r}^{∗}, β_{r}) is the Bockstein spectral sequence for the p-group G(𝔤), then (B_{1}, β_{1}) = (E_{0}, d_{0}) and so B_{2}^{n} = ⊕_{s,t|2s+t=n}E_{1}^{s,t}. Furthermore both B_{∞} and E_{∞} give the cohomology of a point. Thus E_{r}^{∗,∗} can be viewed as an algebraic model for the Bockstein spectral sequence and is computationally useful as we will see. It is known that E_{1} = B_{2}, and the properties of E_{1} will allow us to compute it more readily. (It is unknown whether E_{r} = B_{1+r} for r > 1 but it is true in all computed examples.) In the following we concentrate on computations of B_{2}^{∗} = E_{1}^{∗,∗} = H^{∗}(𝔤, P oly(𝔤^{∗})). The flexibility to change coefficient rings in E_{1}[−]^{∗,∗} will play a very fundamental role, as will the fact that we know the differentials d_{r} in E_{r} raise polynomial degree by r.

The fundamental idea is encoded in the following application of the universal coefficient theorem. This might be a bit abstract here but in the next few sections where explicit examples are worked out it will become clearer. To avoid needless generality we state this theorem for a special case of complex simple Lie algebras though results can easily be extended.

Theorem 6 (Fundamental Comparison Theorem) Let 𝔤 be a ℤ-Lie algebra such that 𝔏 = 𝔤⊗ℂ is a complex simple Lie algebra. (Cartan-Serre basis shows all complex simple Lie algebras arise this way.) Let H^{0}(𝔏, P oly(𝔏^{∗})) = H^{0}(𝔏, U(𝔏)^{∗}) ⊆ P oly(𝔏^{∗}) be denoted by Γ. Then we have

E_{1}[𝔏]^{∗,∗} = H^{∗}(𝔏, U(𝔏)^{∗}) = H^{∗}(𝔏, ℂ) ⊗_{ℂ} Γ

where H^{∗}(𝔏, ℂ) can be identified with the cohomology of a certain (compact form) Lie group G and is an exterior algebra on odd degree generators. (We will see in future sections, that Γ will correspond to H^{∗}(BG, ℂ) which is a polynomial algebra.)

Now since the part of E_{0}[𝔤] corresponding to elements of Hodge degree ≤ N is finitely generated, its cohomology E_{1}[𝔤] can have torsion only for finitely many primes.

Thus for all but a finite number of primes p, we have

dim(E_{1}^{s,t}[𝔤 ⊗ 𝔽_{p}]) = dim(E_{1}^{s,t}[𝔤 ⊗ ℂ])

for all s, t with s ≤ N. On the other hand we will see that if a restriction on Hodge degree is not imposed, this will always break down. For example for 𝔤 = 𝔰𝔩_{2}(ℤ) we will see that E_{1}^{∗,∗}[𝔤] = H^{∗}(𝔤, P oly(𝔤^{∗})) has p-torsion for every prime p and hence is not a finitely generated ring.

Proof The proof of the form of E_{1}[𝔏]^{∗,∗} hinges on two facts.

(1) As 𝔏 is a simple Complex Lie algebra, every finite dimensional complex representation V decomposes as a sum of irreducible representations.

(2) If V is an irreducible representation other than the trivial one-dimensional representation, then H^{∗}(𝔏, V) = 0. (This is called Whitehead’s Lemma, see [GG].) With this we can decompose P oly(𝔏^{∗}) = Γ ⊕ W as U(𝔏)-modules where Γ are the adjoint invariant polynomials; i.e., Γ = H^{0}(𝔏, P oly(𝔏^{∗})), as mentioned in the statement of the theorem. Note W decomposes as a sum of irreducibles, none of which is the trivial one-dimensional representation. Thus

H^{∗}(𝔏, P oly(𝔏^{∗})) = H^{∗}(𝔏, Γ) ⊕ H^{∗}(𝔏, W)

= H^{∗}(𝔏, Γ) by Whitehead’s Lemma

= H^{∗}(𝔏, ℂ) ⊗_{ℂ} Γ as the action of 𝔏 on Γ is trivial.

For every simple complex Lie algebra 𝔏, there exists a real Lie algebra 𝔏_{ℜ} whose complexification is 𝔏 and which is the Lie algebra of a compact connected Lie group G. It is known classically that H^{∗}(𝔏_{ℜ}, ℝ) = H^{∗}(G, ℝ) is an exterior algebra on odd generators (as it is a Hopf algebra). Thus its complexification H^{∗}(𝔏, ℂ) is also an exterior algebra on odd generators.

The final parts are a direct application of the universal coefficient theorem to the complex (E_{0}^{∗,∗}(𝔤), d_{0}) and will be left to the reader. The example for 𝔰𝔩_{2}(ℤ) will be discussed later in the paper.

In the next section we compute the spectral sequence E_{r}^{∗,∗}(𝔰𝔩_{2}(ℂ)) completely using two separate methods, one using just properties of the spectral sequence itself and the second using classical invariant theory for Lie groups. Both recover the classical calculation of Harish-Chandra of the Casimir algebra for 𝔰𝔩_{2}(ℂ). The invariant theory viewpoint provides a concrete meaning to the answer. Then in section 4, we compute the behaviour of the spectral sequence E_{r}^{∗,∗}(𝔏) for any complex simple Lie algebra and relate the answer using invariant theory.

Using the Fundamental Comparison Theorem above we then will get results for the corresponding Lie algebras of corresponding p-groups which we will use to understand higher torsion in their integral cohomology.

3 Computation of Dual Casimirs and Invariant Theory

For a good background discussion of the theory of complex simple Lie algebras and corresponding Lie groups that we use in this paper, see [FH].

Recall 𝔰𝔩_{2}(ℂ) is a Lie algebra of dimension 3 with ℂ-basis

\{x_{h} = [\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}] , x_{e} = [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}] , x_{f} = [\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}]\}

and commutation relations

[x_{h}, x_{e}] = 2x_{e}, [x_{h}, x_{f}] = −2x_{f}, [x_{e}, x_{f}] = x_{h}.

Let h, e, f denote the corresponding dual basis in Λ^{1}(𝔏^{∗}) and let H, E, F denote the “suspended” dual basis in P oly^{1}(𝔏^{∗}).

Thus

E_{0}^{∗,∗}[𝔰𝔩_{2}(ℂ)] = Λ^{∗}(h, e, f) ⊗ ℂ[H, E, F]

and using Theorem 6, we find

E_{1}^{∗.∗}[𝔰𝔩_{2}(ℂ)] = Λ^{∗}(u) ⊗ Γ

where Λ^{∗}(u) = H^{∗}(𝔰𝔩_{2}(ℂ), ℂ) with u = hef (a simple computation) and Γ = H^{0}(𝔰𝔩_{2}(ℂ), U(𝔰𝔩_{2}(ℂ))^{∗}) is the dual Casimir algebra that we seek to find.

It is not hard to see that d_{r} = 0 on Γ for r ≥ 1 as these differentials lower exterior degree and so Γ consists of permanent cycles in the spectral sequence E_{r}^{∗,∗}[𝔰𝔩_{2}(ℂ)]. However, we also know that E_{∞}^{∗,∗}[𝔰𝔩_{2}(ℂ)] is the cohomology of a point, so everything needs to be killed off, thus u must support some differential. Note that H^{∗}(𝔏, P oly^{1}(ad^{∗})) = 0 by Whitehead’s lemma as P oly^{1}(ad^{∗}) = ad^{∗} is a nontrivial irreducible representation of 𝔏. Thus d_{1}(u) = 0 and so E_{1} = E_{2}. Now since the dimension of the Lie algebra is three, and d_{2} lowers exterior degree by three, this is the final possible nonzero differential and so E_{3} = E_{∞} and so we know (E_{2}, d_{2}) is an acyclic complex.

Let κ_{2} = d_{2}(u) then κ_{2} ∈ Γ is an invariant quadratic polynomial. The E_{2} page is concentrated on two horizontal lines: a line with exterior degree 3 with term u ⊗ Γ and a line with exterior degree 0 with term 1 ⊗ Γ. The differential d_{2} between these two lines is given by multiplication by κ_{2}· : Γ → Γ. Since the complex is acyclic we conclude that κ_{2}· : Γ → Γ^{+} is an isomorphism where Γ^{+} are the elements of Γ of positive degree. A simple induction then shows Γ ≅ ℂ[κ_{2}]. This yields the following theorem:

Theorem 7 E_{r}[𝔰𝔩_{2}(ℂ)]-computation:

E_{1}^{∗,∗}[𝔰𝔩_{2}(ℂ)] ≅ H^{∗}(𝔰𝔩_{2}(ℂ), U(sl_{2}(ℂ))^{∗}) ≅ Λ^{∗}(u) ⊗ ℂ[κ_{2}]

where u ∈ E_{1}^{0,3} and κ_{2} ∈ E_{1}^{2,0}. The element κ_{2} is a permanent cycle in the spectral sequence and d_{2}(u) = κ_{2}. κ_{2} is a nonzero multiple of the Killing form of 𝔰𝔩_{2}(ℂ) and hence is a nonzero multiple of H^{2} + EF.

Proof Everything besides the last comment has been already proven. Note for any semisimple complex Lie algebra 𝔤, the Killing form

κ(A, B) = Trace(ad(A) ∘ ad(B) : 𝔤 → 𝔤)

is a symmetric 2-form on 𝔤 which is nondegenerate and hence non-zero. Furthermore it has the property that κ([A, B], C) + κ(B, [A, C]) = 0 for all A, B, C ∈ 𝔤 which implies that it represents a nonzero adjoint-invariant element in

H^{0}(𝔤, P oly^{2}(𝔤^{∗})) = E_{1}^{2,0}[𝔤].

Since in our case H^{0}(𝔤, P oly^{2}(𝔤^{∗})) is one-dimensional and spanned by κ_{2}, the final comment follows. A simple computation shows that the Killing form is a nonzero multiple of H^{2} + EF when the standard identification of quadratic forms with symmetric inner products is made.

A few things to note about the last calculation:

(1) The number of exterior/polynomial generators in E_{1} = H^{∗}(𝔰𝔩_{2}, U(𝔰𝔩_{2})^{∗}) is one each which is equal to the rank of 𝔰𝔩_{2}(ℂ), i.e., the dimension of a Cartan subalgebra in this case the span of x_{h}. This will hold in general for any classical complex simple Lie algebra.

(2) The computation of the dual Casimirs can be used to find the Casimirs, i.e., the central elements in the non-commutative algebra U(𝔰𝔩_{2}). This is because the Killing form sets up an isomorphism between the representations ad and ad^{∗} and hence between H^{0}(𝔰𝔩_{2}, U(𝔰𝔩_{2}(ℂ))) and H^{0}(𝔰𝔩_{2}, U(𝔰𝔩_{2}(ℂ))^{∗}) i.e., between the Casimir algebra and the dual Casimir algebra. One has to be a bit careful though as U(𝔰𝔩_{2}) is only a filtered module and it is noncommutative: the Killing form isomorphism is only between the associated graded of U(𝔤) and U(𝔤)^{∗} in general.

What this means is that when finding the corresponding central elements in U(𝔰𝔩_{2}), one has to do a few things:

(a) Find the duals under the Killing form and view them in the associated graded of U(𝔰𝔩_{2}). Simple computations show κ(x_{e}, x_{f}) = 4 = κ(x_{f}, x_{e}), κ(x_{h}, x_{h}) = 8 and all other inner products of the three basis elements with respect to the Killing form are zero. Thus with respect to the Killing form, the dual of H is \frac{1}{8}x_{h}, the dual of E is \frac{1}{4}x_{f} and the dual of F is \frac{1}{4}x_{e}. Thus the dual of H^{2} + EF ∈ H^{0}(𝔰𝔩_{2}, U(𝔰𝔩_{2})^{∗}) is \frac{1}{64}x_{h}^{2} + \frac{1}{16}x_{f}x_{e} or up to scaling, x_{h}^{2} + 4x_{f}x_{e} ∈ H^{0}(𝔰𝔩_{2}, A) where A is the associated graded of the noncommutative filtered algebra U(𝔰𝔩_{2}).

(b) Find the correct invariant lift from the symmetric algebra A to the noncommutative universal enveloping algebra U(𝔰𝔩_{2}). An invariant lift, i.e., a lift to a central element of U(𝔰𝔩_{2}) is always possible by semisimplicity and this procedure works in general for any complex semisimple Lie algebra. In this case, the correct lift can be obtained by symmetry considerations as x_{h}^{2}+2x_{f}x_{e}+2x_{e}x_{f}.

The physical importance of this computation is as follows. In angular momentum or spin systems, x_{h}, x_{e}, x_{f} represent L_{z}, L_{+} = L_{x}+iL_{y}, L_{−} = L_{x}−iL_{y} respectively, angular momentum operators in 3 directions in ℝ^{3}. Thus their span 𝔰𝔩_{2}(ℂ) represents a space of angular momentum operators taken with respect to all the possible axis directions. In quantum mechanics, a simultaneous measurement of two quantities can be taken if and only if their operators commute. Thus in this case since 𝔰𝔩_{2}(ℂ) has rank one, one can only measure one of these quantities at a time and cannot simultaneously measure say L_{z} and L_{x}. Since it is beneficial to be able to simultaneously measure as many observables as possible to nail down properties of the system, one then is asked if there are any operators in the operator algebra generated by L_{z}, L_{x}, L_{y} which commute with all these axis-specific angular momentum operators. Since U(𝔰𝔩_{2}) is exactly this operator algebra, we find that mathematically we are asking for exactly the center of U(𝔰𝔩_{2}) i.e., the Casimir algebra.

By this calculation, we see the only such operators are polynomials in x_{h}^{2}+2x_{e}x_{f} +2x_{f}x_{e}, which after identifications comes out to be a scalar multiple of the total angular momentum squared L^{2} = L_{x}^{2} + L_{y}^{2} + L_{z}^{2}. Thus in spin systems one can measure L^{2} and, say, L_{z} simultaneously and the corresponding values turn out to be crucial in the physics and chemistry of spin systems. In general we will not comment anymore about finding Casimirs from dual Casimirs as it is not the primary aim of this paper.

While it is possible to perform a similar calculation of the spectral sequence for 𝔰𝔩_{3}(ℂ) purely using properties of the spectral sequence E_{r}^{∗,∗}, for 𝔰𝔩_{n}(ℂ), n > 4 ambiguities arise in differentials. Thus to perform the analogous calculation for an arbitrary complex simple Lie algebra, a supplementary approach has to be taken involving invariant theory. We will repeat the calculation for 𝔰𝔩_{2}(ℂ) using this invariant theory approach now before doing the general calculations over ℂ in the next section. This recalculation will give us additional insight into what we are calculating.

Here are the basic observations needed for the invariant theory viewpoint:

(1) Fix a semisimple Lie algebra 𝔏 over ℂ, then P oly^{∗}(𝔏^{∗}) can be naturally identified as the algebra of polynomial functions on 𝔏. (This identification is natural and injective over infinite fields in general, though over finite fields, different polynomials can induce the same function on the underlying vector space.)

(2) If P oly^{∗}(𝔏^{∗}) is given the dual adjoint action, H^{0}(𝔏, P oly(𝔏^{∗})) = Γ can be identified with the ad-invariant polynomial functions on 𝔏.

(3) If G is a connected Lie group with Lie algebra 𝔏 then the image of the exponential map exp : 𝔏 → G generates G. (Indeed a connected Lie group is generated by a small neighborhood of the identity and the image of the exponential map contains such. The exponential map itself need not be onto, indeed the exponential map exp : 𝔰𝔩_{2}(ℂ) → SL_{2}(ℂ) is not onto.) It can then be shown that the Adjoint action of the Lie group G on 𝔏 has exactly the same invariant polynomial functions on it. Thus

H^{0}(G, P oly(𝔏^{∗})) = H^{0}(𝔏, P oly(𝔏^{∗})) = Γ.

(4) It follows from the remarks above that in the case of G = SL_{n}(ℂ) and 𝔏 = 𝔰𝔩_{n}(ℂ), Γ can be identified with the polynomial functions f : 𝔰𝔩_{n}(ℂ) → ℂ that are invariant under SL_{n}(ℂ)-conjugation.

(5) Consider the characteristic polynomial of a matrix,

P_{𝔸}(x) = det(x𝕀 − 𝔸) = x^{n} − σ_{1}(𝔸)x^{n−1} + σ_{2}(𝔸)x^{n−2} − ⋯ + (−1)^{n}σ_{n}(𝔸).

Note that σ_{j} : 𝔤𝔩_{n} → ℂ is a homogeneous polynomial of degree j in the entries of 𝔸 with value equal to the jth elementary symmetric function of the eigenvalues of 𝔸. Thus for example σ_{1} = trace is a linear polynomial in the entries of 𝔸 while σ_{n} = det is a degree n polynomial in the entries of 𝔸. Since the characteristic polynomial of a matrix is unchanged by conjugation/similarity, we conclude that σ_{1}, σ_{2}, …, σ_{n} represent elements in H^{0}(GL_{n}(ℂ), P oly(𝔤𝔩_{n}^{∗})) = H^{0}(𝔤𝔩_{n}, P oly(𝔤𝔩_{n}^{∗})). It is also clear that these elements are algebraically independent polynomials as they restrict to the elementary symmetric functions when restricted as functions over the diagonal matrices, and these are well-known to be algebraically independent.

(6) It is not hard to show that when restricted to functions over 𝔰𝔩_{n}(ℂ), σ_{1} = 0 but σ_{2}, …, σ_{n} remain algebraically independent. (Just restrict to the diagonals of 𝔰𝔩_{n} to check this). Thus we have found a polynomial subalgebra ℂ[σ_{2}, …, σ_{n}] of Γ = H^{0}(𝔰𝔩_{n}, P oly(𝔰𝔩_{n}^{∗})). Comparing with our previous calculation we see that

H^{0}(𝔰𝔩_{2}(ℂ), P oly(𝔰𝔩_{2}(ℂ))^{∗}) = ℂ[κ_{2}] = ℂ[σ_{2}],

and thus the only conjugation-invariant polynomial functions on 𝔰𝔩_{2}(ℂ) are polynomials in the determinant function, which is itself a quadratic function. (In the next section, we will show the analogous result holds for 𝔰𝔩_{n}(ℂ), i.e., that the conjugation invariant polynomials will be a polynomial algebra on σ_{2}, …, σ_{n}.)

(7) From these theoretical considerations, it follows that the Killing form and determinant are both conjugation-invariant homogeneous quadratic functions 𝔰𝔩_{2}(ℂ) → ℂ and, from the computations, must be a nonzero scalar multiple of each other. Of course this can be explicitly checked:

det ([\begin{matrix}a & b \\ c & −a \end{matrix}]) = −a^{2} − bc = −(a^{2} + bc).

Using the same dual basis stated in the beginning of this section, we then see det = −(H^{2} + EF) and so indeed we have checked that det and the Killing quadratic form κ_{2} are scalar multiples of each other as functions on 𝔰𝔩_{2}(ℂ).

4 Computation of E_{r}^{∗,∗}[𝔤] where 𝔤 is a complex simple Lie algebra

In this section all Lie algebras will be over ℂ and so we will suppress that from the notation. Let us start first with the family 𝔰𝔩_{n}. Many of the arguments are similar for other complex simple Lie algebras.

Recall a Cartan subalgebra 𝔥 (maximal abelian Lie subalgebra consisting of semisimple/diagonalizable elements) is given by the diagonal matrices inside 𝔰𝔩_{n}. For a semisimple complex Lie algebra 𝔤, all Cartan subalgebras are Adjoint-conjugate and the dimension of a Cartan subalgebra is called the rank of the Lie algebra. Thus 𝔰𝔩_{n} has rank n − 1. The adjoint action of 𝔥 on 𝔤 is diagonalizable and a simultaneous nonzero eigenvalue function α : 𝔥 → ℂ is called a root with corresponding eigenvector v called a root vector. Thus

[h, v] = α(h)v

for all h ∈ 𝔥. A root α gives a complex linear functional on 𝔥, i.e., α ∈ 𝔥^{∗}.

The vector space 𝔤_{α} = {v ∈ 𝔤|[h, v] = α(h)v for all h ∈ 𝔥} is called the root space corresponding to the root α. It is standard (see section 14.1 of [FH]) that the root spaces are 1-dimensional and 𝔤 = 𝔥 ⊕ (⊕_{α∈R}𝔤_{α}) where R is the set of roots. The Weyl group W (𝔤) is a group generated by reflections which acts on 𝔥 via automorphisms of the ambient Lie algebra 𝔤 which map the subspace 𝔥 back into itself. In the case of 𝔰𝔩_{n}, the Weyl group is the symmetric group on n letters Σ_{n} which acts on the diagonal matrices 𝔥 by permuting the diagonal entries.

The following lemma is basic to the invariant theory approach to computing the dual Casimirs of 𝔤:

Lemma 1 (Invariants Lemma) Let 𝔤 be a complex simple Lie algebra, 𝔥 a Cartan subalgebra, and W (𝔤) the Weyl group of 𝔤. Let

Γ = H^{0}(𝔤, P oly^{∗}(𝔤^{∗})) = H^{0}(𝔤, U(𝔤)^{∗})

be the algebra of adjoint-invariant polynomial functions on 𝔤 i.e., the dual Casimir algebra. Let H^{0}(W (𝔤), P oly^{∗}(𝔥^{∗})) denote the Weyl group invariant polynomial functions on 𝔥. Then there is an algebra monomorphism

θ : Γ → H^{0}(W (𝔤), P oly^{∗}(𝔥^{∗})),

and furthermore H^{0}(W (𝔤), P oly^{∗}(𝔥^{∗})) can be identified as H^{∗}(BG, ℂ), where G is the classifying space of a simply connected, compact Lie group G corresponding to a compact form 𝔤_{ℝ} of 𝔤.

Proof In this proof, note that a polynomial function on 𝔤 being invariant under the adjoint action of 𝔤 is the same as it being invariant under the Adjoint action of G where G is a connected Lie group having Lie algebra 𝔤 (though the meaning of invariant is slightly different for Lie algebras versus Lie groups). We will capitalize Adjoint when we are specifically referring to the Lie group Adjoint action.

Restriction of adjoint-invariant polynomial functions on 𝔤 to polynomial functions on 𝔥 gives an algebra homomorphism

θ : Γ → H^{0}(W (𝔤), P oly^{∗}(𝔥^{∗}))

as the restriction of an adjoint-invariant polynomial to 𝔥 will yield a polynomial function on 𝔥 which is invariant under the Weyl group (this is because each element of the Weyl group is induced by an Adjoint automorphism of 𝔤). Thus it remains to show injectivity. If f is a nonzero element of Γ then f(v) ≠ 0 for some v ∈ 𝔤. Since f : 𝔤 → ℂ is given by a polynomial, it is continuous. Thus using Jordan decompositions, we can find a semisimple (diagonalizable) element w close to v such that f(w) ≠ 0. (Matrices with distinct eigenvalues form an open dense subset of 𝔰𝔩_{n} for example.) Then w lies in a Cartan subalgebra 𝔥′ of 𝔤 and since any two Cartan subalgebras are conjugate, there is an Adjoint Lie algebra automorphism taking 𝔥′ → 𝔥 which takes w to some element z ∈ 𝔥. By invariance under Adjoint automorphisms, f(w) = f(z) ≠ 0 and so f|_{𝔥} ≠ 0 and so θ is a monomorphism.

The final statement is not needed for any of our p-group results but provides a picture for the complex results so we will only sketch it. If G is a compact, simply connected Lie group G corresponding to a compact form 𝔤_{ℝ} of 𝔤, then a maximal torus T corresponds to a compact form for the Cartan subalgebra 𝔥 and N_{G}(T)/T corresponds to the Weyl group W.

Thus

H^{0}(W, P oly^{∗}(𝔥)^{∗}) = H^{0}(W, H^{∗}(BT, ℂ)) = H^{∗}(BG, ℂ).

The final equality is a well-known result of Borel and will not be reproven here. In the middle we used that H^{∗}(BT, ℂ) can be identified with a polynomial algebra on H^{2}(BT, ℂ) = H^{1}(T, ℂ) = H^{1}(𝔥, ℂ) = 𝔥^{∗}.

As there are a lot of different important equivalences going on, let us look at the explicit case 𝔰𝔩_{n} first. In this case note that a compact form is given by the real Lie algebra 𝔰𝔲_{n} (see section 26.1 in [FH]) with corresponding compact, simply connected Lie group G = SU(n). One has H^{∗}(BSU(n), ℂ) = ℂ[c_{2}, c_{3}, …, c_{n}] a polynomial algebra on the universal Chern classes with deg(c_{i}) = 2i and θ(Γ) ⊆ ℂ[c_{2}, c_{3}, …, c_{n}].

In the last section we saw that ℂ[σ_{2}, σ_{3}, …, σ_{n}] ⊆ Γ with deg(σ_{i}) = 2i. (σ_{i} is a degree i polynomial on 𝔰𝔩_{n} but in our setup as an element in H^{0}(𝔰𝔩_{n}, P oly^{∗}(𝔰𝔩_{n})^{∗}) is an element of total degree 2i due to the suspension of the polynomial variables.) Comparing these set inclusions of graded algebras we conclude:

Theorem 8 (𝔰𝔩_{n} invariant calculation) Let Γ = H^{0}(𝔰𝔩_{n}, P oly^{∗}(ad^{∗})) = H^{0}(𝔰𝔩_{n}, U(𝔰𝔩_{n})^{∗}) be the algebra of adjoint-invariant polynomial functions on 𝔰𝔩_{n}, or equivalently the dual Casimir algebra. Then Γ = ℂ[σ_{2}, σ_{3}, …, σ_{n}] where the σ_{i} have degree 2i and express the ith elementary symmetric polynomial of the eigenvalues of a matrix in terms of the entries of the matrix.

Restriction defines an isomorphism

θ : H^{0}(𝔰𝔩_{n}, U(𝔰𝔩_{n})^{∗}) → H^{0}(Σ_{n}, P oly^{∗}(𝔥^{∗})) = H^{∗}(BSU(n), ℂ) = ℂ[c_{2}, c_{3}, …, c_{n}].

Thus in this case the dual Casimirs of 𝔰𝔩_{n}(ℂ) can be identified with the polynomial algebra on the universal Chern classes c_{2}, c_{3}, …, c_{n}. Finally

E_{1}^{∗,∗}[𝔰𝔩_{n}] = H^{∗}(SU(n) × BSU(n)) = Λ^{∗}(u_{3}, u_{5}, …, u_{2n−1}) ⊗ ℂ[c_{2}, c_{3}, …, c_{n}]

where u_{i} ∈ E_{1}^{0,i} and c_{i} ∈ E_{1}^{i,0}.

It is now a simple matter to work out the full behaviour of the spectral sequence E_{r}^{∗,∗}. We will not really use this for any p-group results so we will be brief. We recall the definition of indecomposables in a connected graded algebra:

Definition 1 Let A = ⊕_{n=0}^{∞}A_{n} be a graded algebra and A^{+} be the ideal of positive degree elements. (We’ll assume A_{0} = k is a copy of the base field.)

A^{+} · A^{+} = {∑_{j=1}^{m} a_{j}b_{j}|a_{j}, b_{j} ∈ A^{+}}

is called the ideal of decomposables.

Q = A/(A^{+} · A^{+})

is called the space of indecomposables.

We now make a series of observations:

(1) Let A^{∗} be E_{1}^{∗,∗}[𝔰𝔩_{n}] graded by total degree. Then the space of indecomposables Q is spanned by {u_{3}, u_{5}, …, u_{2n−1}, c_{2}, c_{3}, …, c_{n}}, and furthermore the c_{i} are permanent cycles and algebraically independent.

(2) Since the d_{r} are derivations the image of a decomposable under d_{r} is decomposable.

(3) The spectral sequence converges to the cohomology of a point and for dimensional reasons the only differentials that can be supported between indecomposables are d_{k}(u_{2k−1}) = μ_{k}c_{k} modulo decomposables, where μ_{k} ∈ ℂ−{0} for 2 ≤ k ≤ n. As the original Chern classes were algebraically independent, redefining c_{k}’s as necessary we can assume μ_{k} = 1 for 2 ≤ k ≤ n.

Thus we conclude:

Theorem 9 The spectral sequence E_{r}^{∗,∗}[𝔰𝔩_{n}] is given by

E_{1}^{∗,∗}[𝔰𝔩_{n}] = Λ^{∗}(u_{3}, u_{5}, …, u_{2n−1}) ⊗ ℂ[c_{2}, c_{3}, …, c_{n}]

where u_{i} ∈ E_{1}^{0,i} and c_{i} ∈ E_{1}^{i,0}. The higher differentials are determined by the fact that the c_{i} are permanent cycles and d_{r}(u_{2r−1}) = c_{r} for 2 ≤ r ≤ n. Thus in particular E_{n+1}^{∗,∗} = E_{∞}^{∗,∗} is the cohomology of a point.

If B_{r}^{∗} denotes the Bockstein spectral sequence of the p-group G(𝔰𝔩_{n}(𝔽_{p})), which is the congruence kernel of the reduction SL_{n}(ℤ/p^{3}ℤ) → SL_{n}(𝔽_{p}), then for any Hodge degree N,

B_{2}^{∗} = E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] ∼ Λ^{∗}(u_{3}, u_{5}, …, u_{2n−1}) ⊗ 𝔽_{p}[c_{2}, c_{3}, …, c_{n}]

where ∼ indicates isomorphism of complexes up to Hodge degree N for all but finitely many primes (depending on N).

Proof All but the last paragraph follows from the observations before the statement of the theorem. The last paragraph follows from the fundamental comparison theorem 6 discussed in previous sections. In this instance we compare B_{2}^{∗} = E_{1}^{∗,∗}[𝔰𝔩_{n}(𝔽_{p})] to E_{1}^{∗,∗}[𝔰𝔩_{n}(ℂ)] through E_{1}^{∗,∗}[𝔰𝔩_{n}(ℤ)]. Excluding the finite set of primes where torsion occurs in E_{1}^{∗,∗}[𝔰𝔩_{n}(ℤ)] in Hodge degrees ≤ N, we have by Theorem 6 that

dim_{𝔽_{p}}(E_{1}^{s,t}[𝔰𝔩_{n}(𝔽_{p})]) = dim_{ℂ}(E_{1}^{s,t}[𝔰𝔩_{n}(ℂ])

for arbitrary t and s ≤ N.

There is a subtlety regarding the ring structure, as the fundamental comparison theorem just shows that the dimensions match up; there is a divided power issue that needs to be addressed when excluding the finite set of primes. To clarify, we know that A^{∗,∗} = E_{1}^{∗,∗}[𝔰𝔩_{n}(ℤ)] = H^{∗}(E_{0}^{∗,∗}[𝔰𝔩_{n}(ℤ)], d_{0}) modulo torsion is a bigraded abelian group where A^{i,j} is free abelian with rank equal to the dimension (as a complex vector space) of E_{1}^{i,j}[𝔰𝔩_{2}(ℂ)]. Thus one can choose generators in A^{0,i} corresponding to the u_{i} (and call them the same thing) and in A^{i,0} corresponding to the c_{i} (and call them the same thing). However in A^{∗,∗} it is not true in general that a combination u_{3}^{α_{2}} … u_{2n−1}^{α_{n}}c_{2}^{a_{2}} … c_{n}^{a_{n}} (0 ≤ α_{i} ≤ 1, a_{i} nonnegative integers), is a generator in its corresponding group; all that is known is that it will be a nonzero integer times a generator. Thus while this element is part of a basis in E_{1}^{∗,∗}[𝔰𝔩_{n}(ℂ)] it is not necessarily a generator element in E_{1}^{∗,∗}[𝔰𝔩_{n}(ℤ)] but only a nonzero integral multiple v of one. (In the case where more than one of these elements lie in the same location A^{i,j}, v should be replaced by the determinant of the nonsingular matrix which expresses the collection of these elements in A^{i,j} in terms of a ℤ-basis of A^{i,j}.)

Now when we reduce to E_{1}^{∗,∗}[𝔰𝔩_{n}(𝔽_{p})], if the nonzero multiple v mentioned above is a multiple of p, the corresponding product becomes zero (or linearly dependent in the case of more than one element in the same position in the E_{1}-page), while if v is not a multiple of p, then the mod p reduction will be a basis element in the corresponding position in the E_{1}-page. Since we know the complex and mod p dimensions match up thru Hodge degree N, to ensure the correct algebra structure, we just have to avoid all primes p dividing the aforementioned multiples v in the finite number of locations up through Hodge degree N in addition to all the primes where p-torsion occurs in E_{1}^{∗,∗}[𝔰𝔩_{n}(ℤ)] up through the same Hodge degree. Since this collection of primes is a finite set the theorem follows.

We will see a bit more detail on this issue when working out the ring structure of E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] in later sections.

Note we would like to emphasize that it is only known that (E_{0}^{∗,∗}, d_{0}) = (B_{1}^{∗}, β) and so E_{1}^{∗,∗} = B_{2}^{∗} by the work in [BP]. It is not known that the higher differentials agree though often this is forced for dimensional reasons.

As a consequence of the last theorem we see for a generic odd prime p, in the Bockstein spectral sequence for G(𝔰𝔩_{n}(𝔽_{p})), B_{2}^{∗} = E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] looks like Λ^{∗}(u_{3}, u_{5}, …, u_{2n−1}) ⊗ 𝔽_{p}[c_{2}, c_{3}, …, c_{n}] for low Hodge degrees. However, when the Hodge degree approaches p, this will break down. One way to see that this has to happen is to note that for low Hodge degree, E_{1}^{∗,∗}[𝔰𝔩_{n}(𝔽_{p})] = B_{2}^{∗} will look like it has Krull dimension n − 1, the rank of 𝔰𝔩_{n}. However once the Hodge degree passes p, the existence of an invariant polynomial algebra on the p-powers of the original dual basis of 𝔰𝔩_{n}[𝔽_{p}] will exhibit a Krull dimension of n^{2} − 1, the dimension of 𝔰𝔩_{n}.

Thus as we will see explicitly in future sections, in the low dimensions the higher torsion B_{2}^{∗} is governed by a “characteristic zero” contribution which has Krull dimension given by the rank of the underlying Lie algebra. However when the Hodge degree approaches p, a phase transition occurs and B_{2}^{∗} explodes behaving now like an algebra which has Krull dimension given by the dimension of the underlying Lie algebra. This phenomenon occurs for all the simple Lie algebras and we will work it out explicitly in further sections for 𝔰𝔩_{2}.

In the remainder of this section, we run quickly through the other families of simple complex Lie algebras as well as the exceptional ones. We will not put in as many details as for the family 𝔰𝔩_{n}, as that would make the paper unwieldy, and our primary concern is the characteristic p considerations. All of the basic calculations needed can be found in Mimura and Toda’s treatise,

[MT] together with Borel’s result that for a compact Lie group G with maximal torus T we have

H^{∗}(BG, ℚ) ≅ H^{∗}(BT, ℚ)^{W} ,

where W is the Weyl group. Letting 𝔤 and 𝔥 be the corresponding Lie algebra and Cartan subalgebra one then can show as done above for 𝔰𝔩_{n} that

H^{∗}(BG, ℂ) ≅ H^{∗}(BT, ℂ)^{W} ≅ P oly(𝔥^{∗})^{W} ≅ H^{0}(𝔤, P oly(ad^{∗})) = H^{0}(𝔤, U(𝔤)^{∗}).

Thus in all cases for a complex simple Lie algebra 𝔤 with corresponding compact form 𝔤_{ℝ} and compact, connected Lie group G, we have

E_{1}^{∗,∗}[𝔤] ≅ H^{∗}(G, ℂ) ⊗ H^{∗}(BG, ℂ).

However, it is important to note that everything in E_{1}^{∗,∗} is expressed in terms of the Lie algebra 𝔤 and invariant polynomial functions and forms on it. In each case the differentials transgress from the “fiber” H^{∗}(G, ℂ) to the “base” H^{∗}(BG, ℂ) for pretty much the same reasons as in the 𝔰𝔩_{𝔫} case. Thus over the complex numbers, E_{r}^{∗,∗}[𝔤] functions very much like the spectral sequence for the fibration G → EG → BG.

This is why we will refer to E_{r}^{∗,∗}[𝔤] as the classifying spectral sequence of the Lie algebra 𝔤. It provides an algebraic model which can be used even when the Lie algebras are defined over fields of characteristic p.

Now without further ado, we summarize the calculation of this spectral sequence for the rest of the complex simple Lie algebras, in each case this gives a picture of the higher torsion in the Bockstein spectral sequence of corresponding p-groups, before the “char 0 to char p” phase transition.

The family 𝔰𝔭_{2n}(ℂ) can be described as follows after suitable choice of basis (see section 16.1 of [FH]):

𝔰𝔭_{2n}(ℂ) = \{[\begin{matrix}A & B \\ C & D \end{matrix}] |B^{T} = B, C^{T} = C, D = −A^{T} , A, B, C, D ∈ 𝔤𝔩_{n}(ℂ)\} .

It has dimension 2n^{2} + n and rank n with a Cartan subalgebra given by diagonal matrices subject to the constraint D = −A^{T}. The Weyl group in this picture is generated by permutations of the entries of A and D simultaneously and consistently with D = −A^{T} and also by pairwise swaps between diagonal elements in A with their negatives in D. With this, it is not hard to show that Weyl group invariant polynomial functions on 𝔥 restrict to polynomials in the elementary symmetric functions on the n diagonal entries in A together with the condition that they be invariant under negation of any variable, and thus polynomials in S_{2}, S_{4}, … where S_{2i} is the ith elementary symmetric polynomial in the squares of the variables. Note S_{2i} ∈ H^{0}(𝔤, P oly(ad^{∗})) is polynomial of degree 2i and hence in E_{1}^{2i,0}, and of total degree 4i.

It turns out 𝔰𝔬_{2n+1}(ℂ) has the same rank n and Weyl group as 𝔰𝔭_{2n}(ℂ) even though they are not isomorphic as Lie algebras in general. This however means that the two corresponding compact forms, the compact symplectic group Sp(n) for Sp_{2n}(ℂ) and SO(2n + 1) for 𝔰𝔬_{2n+1}(ℂ) have the same cohomology with rational (and hence complex) coefficients. The same thing goes for the cohomology of their classifying spaces by Borel’s theorem. Thus we get:

Theorem 10 (Symplectic and odd orthogonal Lie algebras) Let 𝔤 be either 𝔰𝔬_{2n+1}(ℂ) or 𝔰𝔭_{2n}(ℂ). Then

E_{1}^{∗,∗}[𝔤] ≅ H^{∗}(SO(2n + 1), ℂ) ⊗ H^{∗}(BSO(2n + 1), ℂ)
≅ H^{∗}(Sp(n), ℂ) ⊗ H^{∗}(BSp(n), ℂ)
≅ Λ^{∗}(u_{3}, u_{7}, …, u_{4n−1}) ⊗ ℂ[P_{1}, P_{2}, P_{3}, …, P_{n}],

where u_{i} ∈ E_{1}^{0,i} and P_{i} ∈ E_{1}^{2i,0} correspond to the universal Pontryagin classes and Sp(n) is the compact symplectic group. In the spectral sequence, the P_{i}’s are permanent cycles and we have d_{2r}(u_{4r−1}) = P_{r} for 1 ≤ r ≤ n. Thus E_{2n+1}^{∗,∗} = E_{∞}^{∗,∗} is the cohomology of a point. Note the rank of 𝔰𝔬_{2n+1}(ℂ) and 𝔰𝔭_{2n}(ℂ) is n while the dimension is 2n^{2} + n.

As usual this means that the 2nd term of the Bockstein spectral sequence for the p-groups G(𝔰𝔬_{2n+1}(𝔽_{p})) and G(𝔰𝔭_{2n}(𝔽_{p})) satisfy:

B_{2}^{∗} = E_{1}^{∗,∗}[𝔰𝔭_{2n}(𝔽_{p})] ∼ Λ^{∗}(u_{3}, u_{7}, …, u_{4n−1}) ⊗ 𝔽_{p}[P_{1}, P_{2}, …, P_{n}],

where ∼ means that the two complexes are isomorphic in the range of Hodge degrees ≤ N for all but finitely many primes (that depend on N).

We now will just list the results for the other complex simple Lie algebras, the implications for higher torsion in the corresponding p-groups being understood:

Theorem 11 (Even orthogonal Lie algebras) For n ≥ 4,

\begin{aligned}E_{1}^{∗,∗}[𝔰𝔬_{2n}] & ≅ H^{∗}(SO(2n), ℂ) ⊗ H^{∗}(BSO(2n), ℂ) \\ & ≅ Λ^{∗}(u_{3}, u_{7}, …, u_{4n−5}, v) ⊗ ℂ[P_{1}, P_{2}, P_{3}, …, P_{n−1}, e], \end{aligned}

where u_{i} ∈ E_{1}^{0,i}, v ∈ E_{1}^{0,2n−1}. P_{i} ∈ E_{1}^{2i,0} correspond to the universal Pontryagin classes and e ∈ E_{1}^{n,0} corresponds to the universal Euler class and both are permanent cycles in this spectral sequence. Furthermore we have d_{2r}(u_{4r−1}) = P_{r} for 1 ≤ r ≤ n − 1 and d_{n}(v) = e. Note 𝔰𝔬_{2n} has rank n and dimension 2n^{2} − n.

Theorem 12 (Exceptional simple Lie Algebras)

E_{1}^{∗,∗}[𝔤_{2}] ≅ Λ^{∗}(u_{3}, u_{11}) ⊗ ℂ[K_{2}, K_{6}]
E_{1}^{∗,∗}[𝔣_{4}] ≅ Λ^{∗}(u_{3}, u_{11}, u_{15}, u_{23}) ⊗ ℂ[K_{2}, K_{6}, K_{8}, K_{12}]
E_{1}^{∗,∗}[𝔈_{6}] ≅ Λ^{∗}(u_{3}, u_{9}, u_{11}, u_{15}, u_{17}, u_{23}) ⊗ ℂ[K_{2}, K_{5}, K_{6}, K_{8}, K_{9}, K_{12}]

E_{1}^{∗,∗}[𝔈_{7}] ≅ Λ^{∗}(u_{3}, u_{11}, u_{15}, u_{19}, u_{23}, u_{27}, u_{35}) ⊗ℂ[K_{2}, K_{6}, K_{8}, K_{10}, K_{12}, K_{14}, K_{18}]

E_{1}^{∗,∗}[𝔈_{8}] ≅ Λ^{∗}(u_{3}, u_{15}, u_{23}, u_{27}, u_{35}, u_{39}, u_{47}, u_{59})⊗ℂ[K_{2}, K_{8}, K_{12}, K_{14}, K_{18}, K_{20}, K_{24}, K_{30}]

In all cases, the dimension of the Lie algebra is the sum of the subscripts of the u_{i} ∈ E_{1}^{0,i} in its cohomology. As usual, the dual Casimirs K_{i} ∈ E_{1}^{i,0} represent homogeneous degree i adjoint invariant polynomial functions on the Lie algebra and K_{2} always is the quadratic form corresponding to the Killing form.

The results above cover all the complex simple Lie algebras. More generally a semisimple Lie algebra is a direct sum of simple Lie algebras and is covered by the fact that if 𝔤 = 𝔤_{1} ⊕ 𝔤_{2} then E_{0}^{∗,∗}[𝔤] = E_{0}^{∗,∗}[𝔤_{1}] ⊗ E_{0}^{∗,∗}[𝔤_{2}] as differential graded algebras. Furthermore the complex also decomposes as a tensor product with respect to the commuting differential used to define the spectral sequence for the double complex. Thus one has that

E_{r}^{∗,∗}[𝔤] = E_{r}^{∗,∗}[𝔤_{1}] ⊗ E_{r}^{∗,∗}[𝔤_{2}]

as dga’s for any r ≥ 0, i.e., the spectral sequence of the sum is the tensor product of the spectral sequences of the factors. (These comments in fact work over any base field k.) Thus one can work out the spectral sequence for any complex semisimple Lie algebra from the results in this section. A general complex Lie algebra 𝔏 fits in a short exact sequence

0 → rad(𝔏) → 𝔏 → 𝔤 → 0

where rad(𝔏) is the largest solvable ideal of 𝔏 and 𝔤 is semisimple, so it would remain to treat solvable (and in particular nilpotent) Lie algebras. Generally the cohomology of these is much more complicated, though definitely interesting but we will not consider these much in this paper nor the corresponding p-groups as the treatment would require different methods.

We have shown that for any complex semisimple Lie algebra 𝔤, with compact form 𝔤_{ℝ} corresponding to a compact connected Lie group G, that

E_{1}^{∗,∗}[𝔤] = H^{∗}(𝔤, U(𝔤)^{∗}) = H^{∗}(G × BG, ℂ).

We note finally that it is well known that

H^{∗}(ΛBG, ℂ) ≅ H^{∗}(G × BG, ℂ)

by considering the fibration G ≃ ΩBG → ΛBG → BG.

A quick comment regarding related physics: For 𝔤 a complex semisimple Lie algebra we have seen that the Casimir algebra will be a polynomial algebra on h generators where h is the rank of the Lie algebra. In general this means the h linearly independent, commuting operators represented by the Cartan algebra 𝔥 will commute with an additional h Casimir operators which generate the Casimir algebra. This gives a total of 2h commuting and simultaneously measurable operators coming from the Cartan algebra and the Casimir generators, which form (part of) a good system of variables in the underlying physical system.

Thus for example 𝔰𝔩_{3}(ℂ) which plays a role in quark/anti-quark theory as well as meson/baryon classification will have two variables coming from the Cartan algebra, the corresponding measurements of which will correspond to the root weights described in later sections of this paper and an additional two Casimir generators dual to the Chern classes c_{2}, c_{3}, to give a total of 4 commuting operators from this discussion. For a nice mathematical article on the related physics, see [BH].

5 Exponent theory for finite p-groups

In this section we recall the exponent theory of finite groups. Many of these results were introduced in work of A. Adem and Ian Leary (see [A] and [Le]) while the fact that e_{∞}(P) ≠ e(P) was first shown in [Pk]. We summarize the main results regarding these exponents in this section and include a few more for the category of p-groups we will be dealing with.

In this section let G be a finite group. It is well known that the integral cohomology groups H^{n}(G, ℤ), n > 0 are finite abelian groups and |G| · H^{n}(G, ℤ) = 0 for all n > 0. Let H̄(G) = ⊕_{n=1}^{∞}H^{n}(G, ℤ) denote the total reduced integral cohomology of G. We define the following exponents:

Definition 2 (Exponent Definitions) exp(G) = min{n ≥ 1|g^{n} = e for all g ∈ G}. e_{∞}(G) = min{n ≥ 1|nH̄(G) is a finite set}. e(G) = min{n ≥ 1|nH̄(G) = 0}.

The following propositions are discussed in detail in [Pk] so we will not reproduce proofs here. As mentioned before many of them occur in previous independent work of Adem and Leary. Failure of various propositions for exactly one of e_{∞} or e were shown in [Pk]. We write n|m for “the integer n divides the integer m”.

Proposition 1 (Basic Relationship between Exponents) If P is a finite pgroup then

exp(P) | e_{∞}(P) | e(P) | |P |

and furthermore there are examples of p-groups that show that these four quantities are different in general.

Proposition 2 (Subgroups) Let P_{1} ≤ P_{2} be finite p-groups. Then e_{∞}(P_{1})|e_{∞}(P_{2}), exp(P_{1})|exp(P_{2}) and |P_{1}|||P_{2}|. However e(P_{1}) |̸ e(P_{2}) in general.

Proposition 3 (Cyclic groups and Symmetric Sylow subgroups) Let C be a finite cyclic group and S(m) be the Sylow p-subgroup of the symmetric group on m letters.

(1) exp(C) = e_{∞}(C) = e(C) = |C|

(2) p^{n} = exp(S(p^{n})) = e_{∞}(S(p^{n})) = e(S(p^{n})).

Proposition 4 (Products) If G = G_{1} × ⋯ × G_{n}, then

exp(G) = lcm_{i=1,...,n}{exp(G_{i})},
e_{∞}(G) = lcm_{i=1,...,n}{e_{∞}(G_{i})},
e(G) = lcm_{i=1,...,n}{e(G_{i})}

where lcm stands for least common multiple and can be replaced with max in the case of p-groups. Thus if p^{n} is the largest power of p less than or equal to m, then

p^{n} = exp(S(m)) = e_{∞}(S(m)) = e(S(m)).

Proof The first statement follows from an application of Künneth’s Theorem. The second follows from the structure of Sylow p-subgroups of symmetric groups. More precisely, if the p-adic expansion of m is given by m = a_{0} + a_{1}p + a_{2}p^{2} + ⋯ + a_{n}p^{n} then S(m) is the direct product of a_{0} copies of S_{1}, a_{1} copies of S(p) …, up to a_{n} copies of S(p^{n}).

Theorem 13 (Exponent Characterizations) Let P be a p-group. (1)[Nakayama-Rim] If e_{∞}(P) = 1 then P = {e} and e(P) = 1 also. (2)[Adem] If e_{∞}(P) = p then P is elementary abelian and e(P) = exp(P) = p also. (3)[Pakianathan] For odd primes p, the group G(𝔰𝔩_{2}(𝔽_{p})) has exp(P) = e_{∞}(P) = p^{2} and e(P) = p^{3} while |P | = p^{6}. Thus e_{∞}(P) ≠ e(P) in general.

Proposition 5 (Faithful actions on finite sets) Let P be a finite p-group.

(1) If P has a faithful action on a set of size m and p^{n} is the largest power of p less than or equal to m then e_{∞}(P)|p^{n}. Thus e_{∞}(P) is a lower bound on the size of a set on which P can act faithfully.

(2) If the intersection of all subgroups of P of index p^{n} is trivial then e_{∞}(P)|p^{n}. This statement does not hold with e(P) replacing e_{∞}(P).

Theorem 14 (Browder’s exponent theorem and consequences) Let G be a finite group and X a finite dimensional free G − CW -complex, with homologically trivial G-action. Then

|G| | ∏_{n=2}^{∞} exp(H^{n}(G, H_{n−1}(X))) | e(G)^{s(X)}

where s(X) is the number of positive dimensions in which the integral homology of X is nontrivial.

A corollary of this is if E is an elementary abelian group acting freely and homologically trivially on a product of N equal dimensional spheres then rank(E) ≤ N.

Proof The short and elegant proof of this theorem and corollary can be found in [Br]. The second division comes from the definition of e(G). It is unknown to the authors at this time whether the theorem would hold for e_{∞}(G) but we see no reason why it should.

The two fundamental exponents can be recharacterized in terms of the behaviour of the Bockstein spectral sequence of the p-group P. A discussion of this spectral sequence can be found in many places in the literature and is described for example in [BP]. Recall that when applied to the classifying space of a finite p-group P one has that B_{1}^{∗} = H^{∗}(P, 𝔽_{p}) and β_{1} is the Bockstein.

Nonzero permanent cycles in B_{r} represent elements of order p^{r} or greater in the integral cohomology of P and B_{∞} is the cohomology of a point. Thus one has the following:

Lemma 2 (Bockstein Spectral Sequence Characterization) If P is a finite pgroup, then:

(1) e_{∞}(P) = p^{s} if and only if B_{s+1}^{∗} is the first page of the Bockstein spectral sequence whose total complex is finite.

(2) e(P) = p^{k} if and only if B_{k+1}^{∗} is the first page of the Bockstein spectral sequence whose total complex is concentrated in degree 0. (equivalently is 1dimensional).

Finally for p-groups G(𝔏) corresponding to 𝔽_{p}-Lie algebras 𝔏 as in [BP], where 0 → 𝔏 → G(𝔏) → 𝔏 → 0, we have the following:

Proposition 6 (Exponent bounds for G(𝔏)-p-groups.) Let 𝔏 be a 𝔽_{p} Lie algebra such that the intersection of index p^{β} sub Lie algebras is zero. Then e_{∞}(G(𝔏))|p^{2β}.

Proof In the construction of the p-groups G(𝔏) one has in general that |G(𝔏)| = |𝔏|^{2} and so an index p^{β} sub Lie algebra of 𝔏 gives a corresponding index p^{2β} subgroup of G(𝔏). Since the zero Lie subalgebra corresponds to the trivial group, the rest follows from the previous theorems on faithful group actions.

As the reader can see, these exponents play an interesting role in the cohomology of groups limited only by the difficulty in carrying out complete calculations of the integral cohomology of groups. In some results the integral cohomology exponent e(G) arises while in others the asymptotic exponent e_{∞}(G) arises. Since these are not equal in general it is useful to define the extent to which they differ, and the following concept helps do that.

Definition 3 (Exceptional Torsion elements and Exceptional Dimension) Let P be a finite p-group. By definition e_{∞}(P) · H̄^{∗}(P, ℤ) is a finite graded ring. Let the exceptional dimension of P, denoted by ED(P), be defined as the largest dimension in which e_{∞}(P) · H^{∗}(P, ℤ) is nonzero. Note that ED(P) is always a finite nonnegative integer and is zero if and only if e_{∞}(P) = e(P).

Any element x ∈ H^{∗}(P, ℤ) with 0 < ∗ ≤ ED(P) such that e_{∞}(P) · x ≠ 0 is called an exceptional torsion element. Note if ED(P) > 0 then an exceptional torsion element always exists in H^{ED(P)}(P, ℤ) by definition and no exceptional torsion elements exist in dimensions higher than ED(P), i.e., e_{∞}(P) · ⊕_{n=ED(P)+1}^{∞}H^{n}(P, ℤ) = 0.

There is no bound on exceptional dimension in general, i.e., it can be arbitrarily high as the next example shows:

Theorem 15 Let P be the N-fold direct product of G(𝔰𝔩_{2})’s. Then e_{∞}(P) = p^{2}, e(P) = p^{3}. |P | = p^{6N} and ED(P) = N · ED(G(𝔰𝔩_{2})) > 0. Thus lim_{N→∞} ED(P) = ∞.

Proof Follows from the results in this section together with the fact that e_{∞}(G(𝔰𝔩_{2})) = p^{2} and e(G(𝔰𝔩_{2})) = p^{3} proven in [Pk]. Note that the Bockstein spectral sequence of P is the N-fold tensor product of the Bockstein spectral sequence of G(𝔰𝔩_{2}) (this means at each page it’s the tensor product as dga’s.). Also note that ED(G(𝔰𝔩_{2})) is the maximum nonzero dimension of B_{3}^{∗}(G(𝔰𝔩_{2})) while ED(P) is the maximum nonzero dimension of B_{3}^{∗}(P) = ⊗_{j=1}^{N}B_{3}^{∗}(G(𝔰𝔩_{2})) from which the results for essential dimension follow.

The last example is not so strong in the sense that log_{p}(|P |) has to increase in order to get the increase in ED(P). In the next sections we will study the Bockstein spectral sequence for G(𝔰𝔩_{2}) again more carefully using the machinery of Lie algebras we have mentioned in previous sections. More analysis shows in fact the following stronger facts:

(1) ED(G(𝔰𝔩_{2}(𝔽_{p}))) ≥ 2p − 2 for odd primes p. Thus the essential dimension grows with the prime of definition of G(𝔰𝔩_{2}(𝔽_{p})) and is therefore, in some sense, not even bounded for the fixed “group scheme” G(𝔰𝔩_{2}(−)).

(2) Furthermore we will see that the “exceptional torsion elements correspond to the characteristic zero contribution of the Lie algebra scheme 𝔰𝔩_{2}(−) and has Krull dimension given by the rank of the Lie algebra (scheme). Then in the vicinity of the essential dimension a “char 0 to char p” phase transition occurs and the asymptotic torsion has a larger Krull dimension equal to the dimension of the underlying Lie algebra.

(3) In this sense, the word “exceptional torsion” transition to “asymptotic torsion” can be seen in some situations at least, to correspond to a char 0 to char p phase transition in corresponding Lie algebra schemes.

This should become clearer after a few examples.

6 Non-abelian Lie algebra of dimension 2

We will work out an easy example which will introduce the useful concept of weight stratification of the spectral sequence.

Let 𝔤 be the non-abelian Lie algebra (scheme) of dimension 2 with basis x_{h}, x_{e} and commutator given by [x_{e}, x_{h}] = x_{e}. Note 𝔤 can be defined over any base ring k and we will write 𝔤(k) when we want to make the base ring clear. Let h, e denote the dual basis in 𝔤^{∗} and H, E suspended copies in P oly^{1}(𝔤^{∗}). Then

E_{0}^{∗,∗}[𝔤] = Λ^{∗}(h, e) ⊗ k[H, E]

with differential given by d_{0}(h) = d_{0}(H) = 0 and d_{0}(e) = he, d_{0}(E) = Eh−He. The (anti)commuting differential d_{1} giving rise to the spectral sequence is given by d_{1}(h) = H, d_{1}(e) = E, d_{1}(E) = d_{1}(H) = 0. Furthermore by [BP], (E_{0}, d_{0}) = (H^{∗}(G(𝔤), 𝔽_{p}), β) where β is the Bockstein operator when k = 𝔽_{p}.

Now if we assign h, H to have weight 0 regarded as an integer and e, E to have weight 1, and extend weight to E_{0}^{∗,∗}[𝔤] so that the weight of a product is the sum of the weights of the individual factors, then it is easy to see that d_{0} and d_{1} preserve weight and hence so do all differentials of the spectral sequence. Thus the spectral sequence decomposes into a direct sum of spectral sequences given by isolating terms of specific weight:

E_{r}^{∗,∗}[𝔤] = ⨁_{w∈ℕ} E_{r}^{∗,∗}[w]

We will see this occurs in general in the next section, and in the case of semisimple Lie algebras, the sum will be indexed by the root lattice of the corresponding complex semisimple Lie algebra which will be a free abelian group of rank equal to the rank of the Lie algebra. The weight stratification is overkill in this example, but becomes necessary in more complicated computations.

Let us first look at the weight 0 contribution; since the weights in this example are all nonnegative integers, it is easy to see that

E_{0}^{∗,∗}[0] = Λ^{∗}(h) ⊗ k[H],

with d_{0} = 0 identically and d_{1}(h) = H. Thus E_{0}^{∗,∗}[0] = E_{1}^{∗,∗}[0] and E_{2}^{∗,∗}[0] = E_{∞}^{∗,∗}[0] is the cohomology of a point.

Now consider a nonzero weight n > 0. Then E_{0}^{∗,∗}[n] is a right E_{0}^{∗,∗}[0]- module and since d_{0} is a derivation which vanishes on E_{0}^{∗,∗}[0], we have that

d_{0} : E_{0}^{∗,∗}[n] → E_{0}^{∗,∗}[n]

is a right E_{0}^{∗,∗}[0]-module map. Furthermore E_{0}^{∗,∗}[n] is a free E_{0}^{∗,∗}[0]-module of rank 2 with (ordered) basis { eE^{n−1}, E^{n} }. Thus we can represent d_{0} on this weight n-piece as a 2 × 2 matrix with entries in E_{0}^{∗,∗}[0] = Λ^{∗}(h) ⊗ k[H].

Computing we have:

d_{0}(eE^{n−1}) = (he)E^{n−1} − e(n − 1)E^{n−2}(Eh − He) = eE^{n−1}(−nh)

and

d_{0}(E^{n}) = nE^{n−1}(Eh − He) = eE^{n−1}(−nH) + E^{n}(nh)

and hence

d_{0}(eE^{n−1}γ + E^{n}δ)) = eE^{n−1}γ′ + E^{n}δ′

for any γ, δ ∈ E_{0}^{∗,∗}[0] with

[\begin{matrix}γ′ \\ δ′ \end{matrix}] = [\begin{matrix}−nh & −nH \\ 0 & +nh \end{matrix}] [\begin{matrix}γ \\ δ \end{matrix}] ,

and thus d_{0} : E_{0}^{∗,∗}[n] → E_{0}^{∗,∗}[n] is represented by the matrix [\begin{matrix}−nh & −nH \\ 0 & +nh \end{matrix}]

with respect to the ordered basis mentioned above.

Restricting to the case where k is a field, there are now two cases to consider: Case 1: n = 0 in k. In this case d_{0} = 0 on E_{0}^{∗,∗}[n] and E_{0}^{∗,∗}[n] = E_{1}^{∗,∗}[n]. Direct computation shows d_{1}(eE^{n−1}) = E^{n} and we conclude that E_{2}^{∗,∗}[n] = E_{∞}^{∗,∗}[n] = 0.

Case 2: n ≠ 0 in the field k. Note that a typical element in E_{0}^{∗,∗}[0] = Λ^{∗}(h) ⊗ k[H] can be written in the

form γ_{1} + hγ_{2} where γ_{i} are k-polynomials in H. If [\begin{matrix}γ_{1} + hγ_{2} \\ δ_{1} + hδ_{2} \end{matrix}] is in the kernel

of the matrix then

[\begin{matrix}0 \\ 0 \end{matrix}] = [\begin{matrix}−nh & −nH \\ 0 & nh \end{matrix}] [\begin{matrix}γ_{1} + hγ_{2} \\ δ_{1} + hδ_{2} \end{matrix}] = [\begin{matrix}−nh(γ_{1} + Hδ_{2}) − nHδ_{1} \\ +nhδ_{1} \end{matrix}] .

Since n is a unit in k in this case, it follows quickly that δ_{1} = 0 and γ_{1} +

Hδ_{2} = 0. Thus the typical element in the kernel of d_{0} looks like [\begin{matrix}−Hδ_{2} + hγ_{2} \\ hδ_{2} \end{matrix}].

However such an element is in the image of d_{0} as

[\begin{matrix}−nh & −nH \\ 0 & +nh \end{matrix}] [\begin{matrix}\frac{1}{n}(−γ_{2}) \\ \frac{1}{n}(δ_{2}) \end{matrix}] = [\begin{matrix}−Hδ_{2} + hγ_{2} \\ hδ_{2} \end{matrix}] .

Thus the cohomology of the complex (E_{0}^{∗,∗}[n], d_{0}) is identically zero, i.e., E_{1}^{∗,∗}[n] = E_{∞}^{∗,∗}[n] = 0.

Summarizing to the main fields of interest ℂ and 𝔽_{p} we get:

Theorem 16 (Non-abelian Lie algebra of dimension 2) Let 𝔤 be the k-Lie algebra with basis x_{h}, x_{e} and bracket given by [x_{e}, x_{h}] = x_{e}. Then E_{0}^{∗,∗} = Λ^{∗}(h, e) ⊗ k[H, E]. For k a field of characteristic zero we have

E_{1}^{∗,∗} = H^{∗}(𝔤, U(𝔤^{∗})) = E_{1}^{∗,∗}[0] = Λ^{∗}(h) ⊗ k[H]

and E_{2}^{∗,∗} = E_{∞}^{∗,∗} is the cohomology of a point. Note the nonzero contribution to E_{1}^{∗,∗} is only from weight 0 and that the Krull dimension of E_{1} is the rank of the Lie algebra which is one.

For k a field of characteristic p, we have

E_{1}^{∗,∗} = H^{∗}(𝔤, P oly(𝔤^{∗})) = Λ^{∗}(h, eE^{p−1}) ⊗ k[H, E^{p}]

with d_{1}(h) = H and d_{1}(eE^{p−1}) = E^{p}. Note the nonzero contribution to E_{1}^{∗,∗} is only from weights that are congruent to 0 modulo p. Also the Krull dimension is now two, which is the dimension of the Lie algebra. The phase transition from the characteristic 0 answer first occurs in dimension where eE^{p−1} lies i.e., in total dimension 2p − 1 (Hodge degree p − 1).

Following [BP], this yields the following corollary:

Corollary 2 Let 𝔤 be the nonabelian Lie algebra above defined over 𝔽_{p}, p an odd prime. Let G(𝔤) be the corresponding p-group of order p^{4}. Then E_{r}^{∗,∗} = B_{r+1}^{∗} for all r where B_{r}^{∗} is the Bockstein spectral sequence of the group. Thus

E_{1}^{∗,∗} = B_{2}^{∗} = Λ^{∗}(h, eE^{p−1}) ⊗ 𝔽_{p}[H, E^{p}]

and E_{2}^{∗,∗} = B_{3}^{∗} is the cohomology of a point. Thus e_{∞}(G(𝔤)) = e(G(𝔤)) = p^{2} = exp(G(𝔤)).

Proof As mentioned before it was shown in [BP] that (E_{0}^{∗,∗}, d_{0}) = (B_{1}^{∗}, β) in general and hence that E_{1}^{∗,∗} = B_{2}^{∗}. Using comparisons to the cyclic p^{2}- subgroups generated by the kernel of e and h respectively (note Λ^{1}(h, e) = H^{1}(G(𝔤), 𝔽_{p}) = Hom(G(𝔤), 𝔽_{p})), one sees that one must have β_{2}(h) = H and β_{2}(eE^{p−1}) = E^{p} at least up to nonzero scalars. This shows that B_{3}^{∗} has the cohomology of a point and that the two spectral sequences coincide on all pages. The rest follows immediately.

Even though e = e_{∞} in this example, note that there is still a transition in the higher torsion B_{2}^{∗} from behaving like the characteristic zero contribution (krull dimension one) before Hodge degree p − 1 and then changing to a Krull dimension two behaviour after Hodge degree p. Thus for E_{1}^{∗,∗} the contribution Λ^{∗}(h, H) is universal while the contribution Λ^{∗}(eE^{p−1}) ⊗ k[E^{p}] only occurs in characteristic p.

We will see that this behaviour is relatively generic and occurs also in higher dimensional examples.

7 Weight Stratification of the Spectral Sequence E_{r}^{∗,∗}

Let 𝔤 be any complex semisimple Lie algebra and let 𝔥 be a Cartan subalgebra. The adjoint action of 𝔥 on 𝔤 is (simultaneously) diagonalizable and decomposes 𝔤 as

𝔤 = 𝔥 ⊕ (⊕_{α∈R}𝔤_{α})

where R is the set of roots of the Lie algebra. Recall a root α with its root vector v satisfies

[h, v] = α(h)v

for all h ∈ 𝔥. It turns out the values of α are always integral, and so each root α can be viewed in ℤ^{n} where n is the rank of the Lie algebra, i.e., the dimension of its Cartan subalgebra. The correspondence comes by taking a basis {h_{1}, …, h_{n}} of 𝔥 and sending α to (α(h_{1}), …, α(h_{n})).

The Cartan algebra itself corresponds to v such that [h, v] = 0 = 0v for all h ∈ 𝔥, so we will say it has weight 0 (though 0 is usually not considered a root for various reasons).

For a semisimple Lie algebra the set of roots is always finite, and if α is a root, then so is −α. Furthermore, the sublattice of ℤ^{n} spanned by the roots is of rank n also and is called the root lattice.

We now extend this root decomposition to a weight decomposition of E_{0}^{∗,∗} = Λ^{∗}(𝔤^{∗}) ⊗ P oly(𝔤^{∗}) as follows. Decompose Λ^{1}(𝔤^{∗}) = 𝔤^{∗} = P oly^{1}(𝔤^{∗}) by declaring something to have weight α if it is dual to a (nonzero) element of weight −α in 𝔤. Then extend this weighing to all of E_{0}^{∗,∗} by declaring the weight of a product to be the sum of the weights. It is not hard to see that this decomposes E_{0}^{∗,∗} into a direct sum of “weight subspaces” where the weights are any nonnegative integer combination of the root weights, i.e., are the elements of the root lattice.

Though the minus sign is not really important, the reason the dual of a weight α element is said to have weight −α is that if v ∈ 𝔤 is a root vector, [h, v] = α(h)v for all h ∈ 𝔥, and ϕ ∈ 𝔤^{∗} is a dual functional to v (which vanishes on any root vector of different weight), then in the dual adjoint action,

h · ϕ(s) = ϕ([s, h]) = −ϕ([h, s])

and so we have h · ϕ = −α(h)ϕ for all h ∈ 𝔥. Thus it is reasonable to define ϕ to have weight −α. (However, the weight stratification and all the important properties we need would still hold if we dropped the minus sign.)

Finally, it is well known that [𝔤_{α}, 𝔤_{β}] ⊆ 𝔤_{α+β}. From this it is clear from the definition of d_{0} that it preserves weight as it does on the generating set. Note d_{0}(x_{k}) = − ∑_{i<j}^{n} c_{ij}^{k}x_{i}x_{j} and it is clear that c_{ij}^{k} = 0 unless the weight of x_{k} is the sum of the weights of x_{i} and x_{j}. Since d_{1} also preserves weight trivially, we see since all differentials are derived from d_{0} and d_{1} that the spectral sequence respects the weight decomposition.

Thus letting Λ(R) denote the root lattice we have a direct sum decomposition of spectral sequences

E_{r}^{∗,∗}[𝔤] = ⨁_{α∈Λ(R)} E_{r}^{∗,∗}[α].

Note it is not necessary for a Lie algebra to be semisimple to have a weight decomposition. Indeed the example in the last section is not semisimple. All that is needed is a decomposition of 𝔤 into 𝔤_{α} over some abelian semigroup index R ⊆ ℤ^{n} with [𝔤_{α}, 𝔤_{β}] ⊆ 𝔤_{α+β}. Of course the corresponding weight decomposition of E_{0}^{∗,∗} will use as index set the set of nonnegative integer combinations of R which might not be a lattice if −R ≠ R. This was the case in the example in the last section as we only got a ray and not a lattice in ℤ as our weights.

Over a field of characteristic zero, E_{1}^{∗,∗} = H^{∗}(𝔤, U(𝔤^{∗})) = H^{∗}(𝔤)⊗Γ where Γ are the adjoint invariant polynomial functions on 𝔤. Note that if f is an adjoint invariant polynomial function, then by definition [g, f] = 0 = 0f for all g ∈ 𝔤 and so in particular for all g ∈ 𝔥. Thus an adjoint invariant polynomial is by necessity of weight zero. So Γ ⊆ E_{1}^{∗,∗}[0]. However we have previously seen that the generators in H^{∗}(𝔤) transgress to elements in Γ in this spectral sequence and so they also must have weight 0. Thus we conclude for a complex semisimple Lie algebra, H^{∗}(𝔤, U(𝔤)^{∗}) = E_{1}^{∗,∗} = E_{1}^{∗,∗}[0] and so only the weight zero contributions survive in the E_{1}-page and beyond.

Over a field of characteristic p, it is still true that E_{1}^{∗,0} consists of invariant polynomials and hence must have eigenvalue 0 under the adjoint action of 𝔤 on itself. However since our original weight decomposition used integral weights, we can only conclude that the weight of an invariant polynomial is congruent to 0 modulo p as all those weights will be zero in the field. Furthermore not every element of the spectral sequence transgresses to the line of invariant polynomials in general but those which do will have weight congruent to 0 mod p of course.

We summarize this section’s discussion in:

Theorem 17 (Weight stratification) Let 𝔤 be a complex semisimple Lie algebra. Let 𝔤_{ℤ} be a corresponding integral Lie algebra (always exists by Cartan-Serre basis). We can then get a corresponding lie algebra 𝔤_{k} over any ring of definition k.

The root decomposition of 𝔤 induces a weight decomposition of spectral sequences:

E_{r}^{∗,∗}[𝔤_{k}] = ⨁_{α∈Λ(R)} E_{r}^{∗,∗}[α]

where Λ(R) is the root lattice of 𝔤 and is a free abelian group of rank equal to the rank of 𝔤.

If k is a field of characteristic zero then

E_{1}^{∗,∗}[𝔤_{k}] = E_{1}^{∗,∗}[0] = H^{∗}(𝔤, U(𝔤)^{∗});

i.e., only weight zero terms contribute anything to the E_{1}-page or beyond.

If k is a field of characteristic p then E_{1}^{∗,0} consists of polynomials of weight zero modulo p. Furthermore any elements that transgress to the polynomial line also have to have weight zero modulo p.

Though this decomposition might seem technical, as the example in the last section shows, it helps organize computations very effectively.

8 Witt Lie algebras and Algebraic de Rham Cohomology

Since the computations in prime characteristic will be more difficult by far than in characteristic zero, we will have to bring in the additional important computational viewpoint of algebraic D-modules. We use only the most basic algebraic settings of this theory and recall the essential definitions below, but for an excellent introduction in the basic theory the reader is referred to [Co]. A basic discussion about a large family of infinite semisimple Lie algebras related to the classical Witt Lie algebra can be found in [NP].

Let k be a field and P = k[x_{1}, …, x_{n}] be a polynomial algebra in n variables. Formal differentiation \frac{∂}{∂x_{i}} and multiplication by x_{i} define operators i.e., elements of End_{k}(P, P). The operators \frac{∂}{∂x_{i}} and multiplication operators generate under composition a formal operator algebra on the polynomial algebra called the Weyl algebra in n variables. A typical element is a formal differential operator with polynomial coefficients, e.g. x_{1}x_{2} \frac{∂^{2}}{∂x_{1}∂x_{2}} − x_{3} \frac{∂}{∂x_{2}} .

If we define the usual commutator Lie bracket of operators [ψ, τ] = ψ ∘ τ − τ ∘ ψ then the multiplication operators and formal differentiation operators \frac{∂}{∂x_{i}}, 1 ≤ i ≤ n, can be used to define a Lie subalgebra of the Weyl algebra called the Witt algebra in n variables which will be denoted by 𝕎_{n}(k). The typical element of the Witt Lie algebra looks like ∑_{i=1}^{n} P_{i} \frac{∂}{∂x_{i}} where P_{i} are polynomials, i.e., is a first order formal differential operator. Equivalently the Witt Lie algebra can be thought of as the Lie algebra of polynomial vector fields on affine space k^{n}. The Lie bracket is given by

[P \frac{∂}{∂x_{i}} , Q\frac{∂}{∂x_{j}} ] = (P \frac{∂Q}{∂x_{i}} ) \frac{∂}{∂x_{j}} − (Q\frac{∂P}{∂x_{j}} ) \frac{∂}{∂x_{i}} .

Before we see how the Witt Lie algebra plays a role in our computations, we have to look at the dga (E_{0}^{∗,∗}[𝔤], d_{0}) as an algebraic De Rham complex.

Let

Λ_{DR}^{k}(𝔤) = E_{0}^{∗,k} = Λ^{k}(𝔤^{∗}) ⊗ k[𝔤^{∗}] = Λ^{k}(𝔤^{∗}) ⊗ P,

where we have let P denote the polynomial algebra k[𝔤^{∗}] = P oly(𝔤^{∗}). The differential d_{0} then defines a formal algebraic De Rham complex

0 → Λ_{DR}^{0} → Λ_{DR}^{1} → ⋯ → Λ_{DR}^{n} → 0,

where n is the dimension of 𝔤. Recall elements in P oly(𝔤^{∗}) can be thought of as formal polynomial functions on 𝔤 (formal in the case of finite fields) and so the elements in Λ_{DR}^{m} can be thought of as polynomial m-forms on 𝔤.

Let us call the cohomology of this complex the De Rham cohomology of 𝔤, and denote it H_{DR}^{∗}(𝔤). Note the (De Rham) cohomology is then

H_{DR}^{m}(𝔤) = E_{1}^{∗,m}[𝔤] = H^{m}(𝔤, P oly(𝔤^{∗})).

Although, this is essentially just rephrasing, the viewpoint will be useful when we consider 𝔤 = 𝔰𝔩_{2}(𝔽_{p}), which we do next as an explicit example. Recall in this case E_{0}^{∗}[𝔰𝔩_{2}(𝔽_{p})] = Λ^{∗}(h, e, f) ⊗ 𝔽_{p}[H, E, F] is the cohomology of the p-group G(𝔰𝔩_{2}(𝔽_{p})) and d_{0} is the Bockstein β.

Here we are using the same generators as were used in the previous discussion of 𝔰𝔩_{2}(ℂ). Since we will be using the weight stratification a lot in the computations, we will rename the generators now.

The generators h and H are dual to the Cartan algebra element v_{h} =

[\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}] and so have weight 0. We will denote them by x_{0} and y_{0} respectively

now. The generators e and E are dual to the weight 2 element v_{e} = [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}] and

so have weight −2. We will denote them by x_{−} and y_{−} respectively now. The

generators f and F are dual to the weight −2 element v_{f} = [\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}] and so have

weight +2. We will denote them by x_{+} and y_{+} respectively now.

Thus explicitly H^{∗}(G(𝔰𝔩_{2}(𝔽_{p})), 𝔽_{p}) = E_{0}^{∗,∗} = Λ^{∗}(x_{0}, x_{−}, x_{+})⊗𝔽_{p}[y_{0}, y_{−}, y_{+}]. Since [v_{h}, v_{e}] = 2v_{e}, [v_{h}, v_{f}] = −2v_{f} and [v_{e}, v_{f}] = v_{h}, the (Bockstein) differential d_{0} = β can be obtained from equation 1 in Section 2 and is given by

\begin{aligned}β(x_{0}) & = x_{+}x_{−} \\ β(x_{+}) & = 2x_{0}x_{+} \\ β(x_{−}) & = −2x_{0}x_{−} \\ β(y_{0}) & = y_{−}x_{+} − y_{+}x_{−} \\ β(y_{+}) & = −2y_{0}x_{+} + 2y_{+}x_{0} \\ β(y_{−}) & = 2y_{0}x_{−} − 2y_{−}x_{0} \end{aligned}

Here anything with a subscript of zero has weight zero, a negative subscript has weight −2 and a positive subscript has weight +2. Note the root lattice of 𝔰𝔩_{2} is 2ℤ.

Since 𝔰𝔩_{2}(𝔽_{p}) is 3-dimensional, we can make the following identifications. We can identify Λ_{DR}^{0}(𝔰𝔩_{2}(𝔽_{p})) with the polynomial algebra P = 𝔽_{p}[y_{0}, y_{−}, y_{+}]. We can identify a typical element f_{0}x_{0} + f_{−}x_{−} + f_{+}x_{+} of Λ_{DR}^{1}(𝔰𝔩_{2}(𝔽_{p})) with f̂ = (f_{0}, f_{−}, f_{+}) ∈ P^{3}. We will view f̂ is a vector-valued polynomial function. We can identify a typical element g_{0}x_{−}x_{+}+g_{−}x_{+}x_{0}+g_{+}x_{0}x_{−} of Λ_{DR}^{2}(𝔰𝔩_{2}(𝔽_{p})) with ĝ = (g_{0}, g_{−}, g_{+}) ∈ P^{3}. Finally we can identify a typical element θx_{0}x_{−}x_{+} ∈ Λ_{DR}^{3}(𝔰𝔩_{2}(𝔽_{p})) with θ ∈ P.

We then define the 𝔰𝔩_{2} gradient, curl and divergence, ∇_{𝔰𝔩_{2}}, Curl_{𝔰𝔩_{2}}, ∇_{𝔰𝔩_{2}}· by saying the following diagram commutes:

\begin{matrix}Λ_{DR}^{3} & ⟶ & P \\ ^{β}↑ & & ↑^{∇_{𝔰𝔩_{2}} ·} \\ Λ_{DR}^{2} & ⟶ & P^{3} \\ ^{β}↑ & & ↑^{Curl_{𝔰𝔩_{2}}} \\ Λ_{DR}^{1} & ⟶ & P^{3} \\ ^{β}↑ & & ↑^{∇_{𝔰𝔩_{2}}} \\ Λ_{DR}^{0} & ⟶ & P \end{matrix}  (2)

where the horizontal maps are the identifications mentioned in the preceding paragraph.

For P ∈ Λ_{DR}^{0} we can write β(P) = x_{0}β_{0}(P) + x_{−}β_{−}(P) + x_{+}β_{+}(P) where β_{i} : P → P are easily seen to be derivations. Since the derivations P → P can in general be identified with the Witt Lie algebra, let us find such identifications for β_{0}, β_{−}, β_{+}. Since these derivations are uniquely determined by their action on the generators y_{0}, y_{−}, y_{+} of P, simple computations using the Bockstein formulas listed above show that we have the following identifications:

\begin{matrix}β_{0} = 2y_{+} \frac{∂}{∂y_{+}} − 2y_{−} \frac{∂}{∂y_{−}} \\ β_{−} = −y_{+} \frac{∂}{∂y_{0}} + 2y_{0} \frac{∂}{∂y_{−}} \\ β_{+} = y_{−} \frac{∂}{∂y_{0}} − 2y_{0} \frac{∂}{∂y_{+}} . \end{matrix}  (3)

It is a simple computation to find the operator commutators [β_{0}, β_{−}] = 2β_{−}, [β_{0}, β_{+}] = −2β_{+} and [β_{−}, β_{+}] = β_{0} and hence these operators give a faithful representation of the Lie algebra 𝔰𝔩_{2} in the Witt Lie algebra on 3 generators.

Straightforward computations then express the 𝔰𝔩_{2}-gradient, curl and divergence in terms of these linear differential operators.

\begin{aligned}∇_{𝔰𝔩_{2}}(θ) & = (β_{0}(θ), β_{−}(θ), β_{+}(θ)); \\ Curl_{𝔰𝔩_{2}}(f̂) & = (β_{−}f_{+} − β_{+}f_{−} − f_{0}, β_{+}f_{0} − β_{0}f_{+} − 2f_{+}, β_{0}f_{−} − β_{−}f_{0} − 2f_{−}); \\ ∇_{𝔰𝔩_{2}} · f̂ & = β_{0}f_{0} + β_{−}f_{−} + β_{+}f_{+}, \end{aligned}

for all vector polynomials f̂ = (f_{0}, f_{−}, f_{+}) ∈ P^{3} and scalar polynomials θ ∈ P. Since β ∘ β = 0, one has the usual calculus identities:

\begin{aligned}∇_{𝔰𝔩_{2}} · Curl_{𝔰𝔩_{2}}(f̂) & = 0 and \\ Curl_{𝔰𝔩_{2}}(∇_{𝔰𝔩_{2}}(θ)) & = 0 \end{aligned}

for all f̂ ∈ P^{3} and θ ∈ P. Thus H_{DR}^{3} = H^{3}(𝔰𝔩_{2}, P oly(𝔰𝔩_{2})^{∗}) measures the obstructions to writing a polynomial as a divergence of a vector polynomial, H_{DR}^{2} = H^{2}(𝔰𝔩_{2}, P oly(𝔰𝔩_{2})^{∗}) measures the obstructions to writing a divergencefree polynomial vector as a curl of another polynomial vector, H_{DR}^{1} = H^{1}(𝔰𝔩_{2}, P oly(𝔰𝔩_{2})^{∗}) measures the obstructions to writing a curl-free polynomial vector as the gradient of a polynomial, and H_{DR}^{0} = H^{0}(𝔰𝔩_{2}, P oly(𝔰𝔩_{2})^{∗}) are the polynomials with gradient identically equal to zero. Thus the DeRham cohomology plays exactly the same role as it does in calculus, except that the basic notions of gradient, curl and divergence have been altered in accordance with the Lie algebra 𝔰𝔩_{2}.

The goal now is to compute E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] = H^{∗}(𝔰𝔩_{2}(𝔽_{p}), P oly(𝔰𝔩_{2}(𝔽_{p}))^{∗}) as it gives B_{2}^{∗} in the Bockstein spectral sequence of the p-group G(𝔰𝔩_{2}(𝔽_{p})). This is equivalent to computing H_{DR}^{∗}(𝔰𝔩_{2}(𝔽_{p})) as mentioned above. We have seen general comparison theorems that give the answer in low Hodge degrees relative to the prime p but we seek now to compute the answer more explicitly in order to understand the “char 0 to char p” phase transition.

We record some basic facts about Λ_{DR}^{∗}(𝔰𝔩_{2}) that will be used in our computations in the next section. As mentioned before, we weigh the polynomials in P = 𝔽_{p}[y_{0}, y_{−}, y_{+}] by giving y_{0} weight 0, y_{−} weight −2 and y_{+} weight +2 and declaring the weight of a product to be the sum of the weights.

Proposition 7 (Eigenfunctions of β_{0}) If f ∈ P is a polynomial of weight w, then β_{0}(f) = wf. Furthermore the monomial basis of P is a basis of β_{0}- eigenfunctions.

Proof A simple computation shows that the identity holds for the generators y_{0}, y_{−}, y_{+}. To finish the proof we note that if ϕ_{1} has weight a_{1} and ϕ_{2} has weight a_{2} then as β_{0} is a derivation, we have β_{0}(ϕ_{1}ϕ_{2}) = β_{0}(ϕ_{1})ϕ_{2}+ϕ_{1}β_{0}(ϕ_{2}) = (a_{1} + a_{2})ϕ_{1}ϕ_{2} and the proposition then follows from simple induction and the fact that P has a basis of monomials which are products of generators.

Proposition 8 (Weight of exterior 0 and 3 lines.) H_{DR}^{3} = H^{3}(𝔰𝔩_{2}(𝔽_{p}), P oly(𝔰𝔩_{2}(𝔽_{p}))^{∗}) and H_{DR}^{0} = H^{0}(𝔰𝔩_{2}(𝔽_{p}), P oly(𝔰𝔩_{2}(𝔽_{p}))^{∗}) consist completely of elements of weights congruent to 0 mod p.

The polynomials κ = y_{0}^{2} + y_{+}y_{−}, y_{0}^{p}, y_{−}^{p}, y_{+}^{p} lie in H_{DR}^{0}.

Proof If θ ∈ H_{DR}^{0} − {0} is of weight w then β(θ) = 0 so in particular β_{0}(θ) = wθ = 0. Thus w = 0 in 𝔽_{p}. Since w is regarded as an integer, this is the same as saying w must be congruent to 0 mod p.

Let u = x_{0}x_{−}x_{+}, and let ψu with ψ ∈ P be a typical element in Λ_{DR}^{3}. Suppose the weight w of ψ is not zero modulo p; then β_{0}ψ = wψ ≠ 0 and β_{0}(\frac{ψ}{w}) = ψ. It is then easy to check that ∇_{𝔰𝔩_{2}} · (\frac{ψ}{w} , 0, 0) = ψ and so ψu is a coboundary and represents zero in H_{DR}^{3}. Thus H_{DR}^{3} consists exclusively of elements of weight congruent to 0 mod p as u itself has weight 0.

The final line of the proposition is a trivial computation.

If f_{0}, f_{−}, f_{+} ∈ P have weights m, n, k respectively, we will refer to the weight of the vector polynomial f̂ = (f_{0}, f_{−}, f_{+}) as (m, n, k) ∈ ℤ^{3}.

Proposition 9 (Weight of exterior 1 line.) Let θ = f_{0}x_{0} + f_{−}x_{−} + f_{+}x_{+} ∈ Λ_{DR}^{1} have weight w ∈ ℤ. The corresponding vector f̂ = (f_{0}, f_{−}, f_{+}) has weight (w, w + 2, w − 2). Then we have β(θ) = 0 ↔ Curl_{𝔰𝔩_{2}}(f̂) = 0 if and only if the following equations hold:

\begin{aligned}f_{0} & = β_{−}(f_{+}) − β_{+}(f_{−}) \\ β_{+}(f_{0}) & = wf_{+} \\ β_{−}(f_{0}) & = wf_{−} \end{aligned}

H_{DR}^{1} = H^{1}(𝔰𝔩_{2}(𝔽_{p}), P oly^{∗}(𝔰𝔩_{2}(𝔽_{p})^{∗})) consists completely of elements of weight congruent to 0 mod p.

Proof The equations follow from Curl_{𝔰𝔩_{2}}(f̂) = 0 using the formula for Curl_{𝔰𝔩_{2}} listed above together with simplifications which come from Proposition 7.

Thus it remains to show that H_{DR}^{1} consists only of elements of weight congruent to 0 mod p. Let θ ∈ Λ_{DR}^{1} be an element with β(θ) = 0 and weight w not congruent to 0 mod p. Since w ≠ 0 ∈ 𝔽_{p} we have

∇_{𝔰𝔩_{2}} (\frac{f_{0}}{w} ) = \frac{1}{w} (β_{0}(f_{0}), β_{−}(f_{0}), β_{+}(f_{0})) = \frac{1}{w} (wf_{0}, wf_{−}, wf_{+}) = f̂

where the second to last equality follows from the equations stated in this proposition together with β_{0}(f_{0}) = wf_{0} as f_{0} has weight w. Thus [θ] = 0 ∈ H_{DR}^{1}. So we can conclude that the (nonzero) elements of H_{DR}^{1} all have weight congruent to 0 mod p.

Corollary 3 (Weight of H_{DR}^{∗}) H_{DR}^{∗} = E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] = H^{∗}(𝔰𝔩_{2}(𝔽_{p}), P oly(ad^{∗})) consists entirely of elements of weight 0 mod p. Thus the E_{1} and higher pages of the spectral sequence are concentrated completely in weights 0 modulo p.

Proof From proposition 8 and proposition 9, it follows that any elements of weight not congruent to 0 mod p in E_{1}^{∗,∗} = H_{DR}^{∗} must lie on the exterior 2 line, i.e., on E_{1}^{∗,2}. Since all differentials involving this line either come from or go to elements not on this line, and since we know the spectral sequence respects weight, we conclude that any elements of weight not congruent to 0 mod p on the exterior 2 line would survive until E_{∞}. Since we know E_{∞} is the cohomology of a point, we conclude that all (nonzero) elements on the exterior 2 line must also have weight 0 mod p. Thus the corollary is proven.

Lemma 3 (Weight 0 polynomials) Let V_{0}^{N} denote the vector space of (integral) weight 0, degree N polynomials inside P. The set θ_{ℓ} = y_{0}^{N−2ℓ}(y_{−}y_{+})^{ℓ}, 0 ≤ ℓ ≤ \frac{N}{2} is a basis of V_{0}^{N}. The set Γ_{ℓ} = y_{0}^{N−2ℓ}κ^{ℓ}, 0 ≤ ℓ ≤ \frac{N}{2} is also a basis, where κ = y_{0}^{2} + y_{−}y_{+}.

The algebra P_{0} of integral weight 0 polynomials is 𝔽_{p}[κ, y_{0}].

Proof Any monomial of integral weight zero must involve the same number of y_{−} and y_{+} variables from which it follows the set θ_{ℓ}, 0 ≤ ℓ ≤ \frac{N}{2} is a basis of V_{0}^{N}.

Expanding κ^{ℓ} = (y_{0}^{2} + y_{−}y_{+})^{ℓ}, it is easy to see that Γ_{ℓ} = ∑j=0^{\frac{N}{2}} c_{ℓj}θ_{j} where c_{ℓj} = 0 if j > ℓ and c_{ℓℓ} = 1. Thus the components of Γ_{ℓ} with respect to the basis {θ_{ℓ}} form a lower triangular matrix with 1’s along the diagonal. Since this matrix is invertible, the set {Γ_{j}} is also a basis of V_{0}^{N}. The final line follows from these remarks as κ is algebraically independent from y_{0} since y_{−}y_{+} is.

Lemma 4 (Weight 0 mod p monomials) Let s_{−} = y_{−}^{p} and s_{+} = y_{+}^{p}. If f is a monomial polynomial of weight 0 mod p then f = s_{−}^{k}s_{+}^{ℓ}f_{0} for some nonnegative integers k, ℓ and integral weight 0 polynomial f_{0}.

Proof Follows easily as any monomial with weight 0 mod p, must involve a number of y_{−} variables and a number of y_{+} variables that are congruent modulo p.

Proposition 10 (Euler-Poincare Count Formula) Let k be a field, 𝔤 a finite dimensional k-Lie algebra of dimension n > 0 and M a finite dimensional 𝔤-module. Then

∑_{i=0}^{n}(−1)^{i} dim(H^{i}(𝔤, M)) = 0.

Thus in particular, ∑_{i=0}^{dim(𝔤)}(−1)^{i} dim(E_{1}^{m,i}[𝔤]) = 0 for all nonnegative integers m.

Proof H^{∗}(𝔤, M) can be computed as the cohomology of a Koszul complex of the form Λ^{∗}(𝔤^{∗}) ⊗ M. Letting C^{i} = Λ^{i}(𝔤^{∗}) ⊗ M, one has from the Euler-Poincare lemma:

∑_{i=0}^{n}(−1)^{i} dim(C^{i}) = ∑_{i=0}^{n}(−1)^{i} dim(H^{i}(𝔤, M)).

Since dim(C^{i}) is easily seen to be (\begin{matrix}n \\ i \end{matrix}) dim(M), the alternating sum on the

left comes out to

dim(M) ∑_{i=0}^{n}(−1)^{i} (\begin{matrix}n \\ i \end{matrix}) = dim(M)(1 − 1)^{n} = 0.

The last statement follows as E_{1}^{m,i}[𝔤] = H^{i}(𝔤, P oly^{m}(ad^{∗})).

Proposition 11 (H_{DR}^{0}-computation) Let κ = y_{0}^{2} + y_{−}y_{+}, s_{0} = y_{0}^{p}, s_{−} = y_{−}^{p}, and s_{+} = y_{+}^{p}. Then H_{DR}^{0} = H^{0}(𝔰𝔩_{2}(𝔽_{p}), P oly(ad^{∗})) is generated as an 𝔽_{p}- algebra by κ, s_{0}, s_{+},and s_{−}. Furthermore, s_{0}, s_{−},and s_{+} generate a polynomial algebra and κ is integral over this polynomial algebra satisfying minimal relation κ^{p} = s_{0}^{2} + s_{−}s_{+}.

Proof Let A ⊆ 𝔽_{p}[y_{0}, y_{−}, y_{+}] be the 𝔽_{p}-algebra generated by κ, s_{0}, s_{−}, s_{+}. A ⊆ H_{DR}^{0} by proposition 8. Also we have seen that H_{DR}^{0} is a subset of the weight 0 mod p polynomials which lie in the 𝔽_{p}-algebra generated by y_{0} and A. Let f ∈ H_{DR}^{0} have degree N. Then we can write

f = ∑_{i=0}^{N} y_{0}^{i}P_{i} = ∑_{i=0}^{p−1} y_{0}^{i}Q_{i},

where P_{i}, Q_{i} ∈ A for all i. (We used that y_{0}^{p} = s_{0} ∈ A in the second equality.) Thus f = Q_{0} + ∑_{i=1}^{p−1} y_{0}^{i}Q_{i}. Since β(f) = β(Q_{i}) = 0, we have

0 = ∑_{i=1}^{p−1} β_{+}(y_{0}^{i})Q_{i} = ∑_{i=1}^{p−1} iy_{0}^{i−1}y_{−}Q_{i},

and so ∑_{i=1}^{p−1} iy_{0}^{i}Q_{i} = 0, where in the last step we scaled by y_{0} and used that y_{−} is not a zero divisor in 𝔽_{p}[y_{0}, y_{−}, y_{+}] to cancel it. If G = ∑_{i=1}^{p−1} y_{0}^{i}Q_{i} = f − Q_{0}, then we have seen that since β(G) = 0, ∑_{i=1}^{p−1} iy_{0}^{i}Q_{i} = 0. Since the numbers 0 < i < p are nonzero in 𝔽_{p}, we can use this relation to solve for y_{0}^{p−1}Q_{p−1} as a linear combination of y_{0}^{i}Q_{i} for 0 < i < p − 1. Thus G = ∑_{i=1}^{p−2} y_{0}^{i}T_{i} for new elements T_{i} ∈ A. We can now repeat the argument until we see that G = y_{0}T for some T ∈ A. However β_{+}(G) = 0 = y_{−}T then gives T = 0 and hence G = 0. Thus f = Q_{0} + G = Q_{0} ∈ A and hence we have shown that H_{DR}^{0} ⊆ A. Thus A = H_{DR}^{0}.

Finally s_{0}, s_{−}, s_{+} are algebraically independent as they are the image of the algebraically independent elements y_{0}, y_{−}, y_{+} under the injective Frobenius algebra endomorphism of 𝔽_{p}[y_{0}, y_{−}, y_{+}] given by θ(α) = α^{p}. Since κ = y_{0}^{2} + y_{−}y_{+}, the Frobenius endomorphism also shows that κ^{p} = s_{0}^{2} +s_{−}s_{+} and hence κ is integral over 𝔽_{p}[s_{0}, s_{−}, s_{+}] as it satisfies the monic polynomial t^{p} − (s_{0}^{2} + s_{−}s_{+}). It is clear that this is the minimal polynomial as κ has degree 2 and the s_{i} have degree p, and p is an odd prime.

Proposition 12 (H_{DR}^{3}-computation) Let u ∈ E_{1}^{0,3} and τ ∈ E_{1}^{p−1,3} be βcohomology classes representing x_{0}x_{−}x_{+} and y_{0}^{p−1}x_{0}x_{−}x_{+} respectively. Then u, τ ≠ 0 and E_{1}^{∗,3} = H_{DR}^{3} is generated as a E_{1}^{∗,0} = H_{DR}^{0}-module by τ and u. Furthermore we have the relations us_{+} = us_{−} = us_{0} = 0 and the fundamental relation κ^{\frac{p−1}{2}} u = 0.

This implies τH_{DR}^{0} ∩ uH_{DR}^{0} = 0 and so H_{DR}^{3} = τH_{DR}^{0} ⊕ uH_{DR}^{0}.

Proof By proposition 8, any element of H_{DR}^{3} has weight congruent to 0 mod p. Since x_{0}x_{−}x_{+} has weight 0, such an element has to be represented by x_{0}x_{−}x_{+}F where F is a polynomial of weight 0 mod p. Thus F is a linear combination of things of the form κ^{ℓ}y_{0}^{j}s_{0}^{a}s_{−}^{b}s_{+}^{c} for ℓ, j, a, b, c nonnegative integers and 0 ≤ j ≤ p − 1. We can write F as F = ∑_{i=0}^{p−1} f_{i}y_{0}^{i} where f_{i} ∈ H_{DR}^{0}. (Note β is a H_{DR}^{0}-module map so the elements f_{i} of gradient identically zero function as “constants”.) Thus we see H_{DR}^{3} is generated as a H_{DR}^{0}-module by [x_{0}x_{−}x_{+}y_{0}^{i}] 0 ≤ i ≤ p − 1, where [−] means “cohomology class represented by”. Note u = [x_{0}x_{−}x_{+}y_{0}^{0}] and τ = [x_{0}x_{−}x_{+}y_{0}^{p−1}].

Let us write F_{1} ∼ F_{2} if they differ by ∇_{𝔰𝔩_{2}} ·Ĝ for some Ĝ ∈ P^{3}. In this case x_{0}x_{−}x_{+}F_{1} and x_{0}x_{−}x_{+}F_{2} are β-cohomologous and so uF_{1} = uF_{2} ∈ H_{DR}^{3}.

Let us fix a weight w which is 0 mod p for F and ask about the image of ∇_{𝔰𝔩_{2}}· in this weight w component. To hit this weight component, the input Ĝ = (g_{0}, g_{−}, g_{+}) has to have weight (w, w − 2, w + 2) and the corresponding output will be

β_{0}g_{0} + β_{−}(g_{−}) + β_{+}(g_{+}) = β_{−}(g_{−}) + β_{+}(g_{+}),

as β_{0}g_{0} = 0. Since g_{−} has weight w − 2 congruent to −2 mod p, it is a combination of things of the form y_{−}y_{0}^{i}α or y_{+}^{p−1}y_{0}^{i}α with α ∈ H_{DR}^{0} and 0 ≤ i ≤ p − 1. Since g_{+} has weight w + 2 congruent to +2 mod p, it is a combination of things of the form y_{+}y_{0}^{i}α or y_{−}^{p−1}y_{0}^{i}α, α ∈ H_{DR}^{0}, 0 ≤ i ≤ p − 1. Thus, as g_{−} and g_{+} can be chosen independently, we see that the weight w component of the image of the divergence ∇_{𝔰𝔩_{2}}· is the H_{DR}^{0}-module spanned by the elements β_{+}(y_{+}y_{0}^{i}), β_{+}(y_{−}^{p−1}y_{0}^{i}), β_{−}(y_{−}y_{0}^{i}), β_{−}(y_{+}^{p−1}y_{0}^{i}) with 0 ≤ i ≤ p − 1 (or equivalently for all nonnegative integers i as y_{0}^{p} = s_{0} ∈ H_{DR}^{0}).

After simple computations we find that the image of the divergence is generated as a H_{DR}^{0}-module by the elements iκy_{0}^{i−1} −(i+2)y_{0}^{i+1}, s_{−}iy_{0}^{i−1}, s_{+}iy_{0}^{i−1} for 0 ≤ i ≤ p − 1 (or equivalently all nonnegative i).

Setting θ_{j} = jy_{0}^{j−1} we find that θ_{j} = 0 if j is congruent to 0 mod p and \frac{θ_{j}}{j} = y_{0}^{j−1} if j is not congruent to 0 mod p. In this new language the image of the divergence can be computed as the H_{DR}^{0}-module spanned by κθ_{j} − θ_{j+2}, s_{+}θ_{j}, s_{−}θ_{j} for all nonnegative integers j. Thus κθ_{j} ∼ θ_{j+2} for all nonnegative j, and by a simple induction we have that all θ_{2k} ∼ κ^{k}θ_{0} = 0 and θ_{2k+1} ∼ κ^{k}θ_{1} = κ^{k}.

Thus u[θ_{2k}] = 0 and u[θ_{2k+1}] = uκ^{k} in H_{DR}^{3}. Since θ_{i+1} are nonzero multiples of y_{0}^{i} when 1 ≤ i < p − 1 it follows that u[y_{0}^{i}] = 0 when i is odd in this range and u[y_{0}^{2k}] = uκ^{k} when i = 2k in this range. Since we had previously seen that H_{DR}^{3} is generated by u[y_{0}]^{i} for 0 ≤ i ≤ p−1 as H_{DR}^{0}-module, we now see that all these generators except u[y_{0}^{0}] = u and u[y_{0}^{p−1}] = τ are redundant.

Hence we have shown that H_{DR}^{3} is generated by u and τ as a H_{DR}^{0}-module. Since 1 is not in the image of the divergence, u ≠ 0. Since y_{0}^{p−1} is not in the image of the divergence, τ ≠ 0. Since s_{+}θ_{1} = s_{+}, s_{−}θ_{1} = s_{−} are in the image of the divergence, s_{+}u = s_{−}u = 0. Since y_{0}^{p} = θ_{p+1} ∼ κ^{\frac{p+1}{2}} θ_{0} = 0, we also have s_{0}u = 0. Finally since 0 = θ_{p} ∼ κ^{\frac{p−1}{2}} θ_{1} = κ^{\frac{p−1}{2}} we have κ^{\frac{p−1}{2}} u = 0.

Note the relations that we just found imply that uH_{DR}^{0} is concentrated in Hodge degrees less than or equal to p − 2 as u, uκ, …, uκ^{\frac{p−1}{2} −1} are. On the other hand τH_{DR}^{0} is concentrated in Hodge degrees greater than or equal to p − 1, the Hodge degree of τ. Thus τH_{DR}^{0} ∩ uH_{DR}^{0} = 0 and the proof of the proposition is complete.

Since it is important to know that κ^{i}u ≠ 0 for 0 ≤ i < \frac{p−1}{2} in order to get lower bounds on essential dimension, we show that with a dedicated proof for clarity:

Proposition 13 (Fundamental nonvanishing) In H_{DR}^{3} we have

κ^{\frac{p−3}{2}} u ≠ 0.

Proof To ease notation, we set i = \frac{p−3}{2}. Assume to the contrary that there is an element x_{0}x_{+}F + x_{0}x_{−}G + x_{+}x_{−}H with

β(x_{0}x_{+}F + x_{0}x_{−}G + x_{+}x_{−}H) = κ^{i}u,

where F, G, and H are polynomials of degree p − 3. Then F must have weight −2, and hence is a linear combination of terms of the form y_{0}^{2j+1}y_{+}^{i−j−1}y_{−}^{i−j} for 0 ≤ j < i. Similarly, G is a linear combination of terms of the form y_{0}^{2j+1}y_{+}^{i−j}y_{−}^{i−j−1}, and H is a linear combination of terms of the form y_{0}^{2j}y_{+}^{i−j}y_{−}^{i−j}. A straightforward induction shows that

β(x_{0}x_{+}y_{0}^{k}y_{+}^{ℓ−1}y_{−}^{ℓ}) = x_{0}x_{+}x_{−}(2ℓy_{0}^{k+1}y_{+}^{ℓ−1}y_{−}^{ℓ−1} − ky_{0}^{k−1}y_{+}^{ℓ}y_{−}^{ℓ})
= β(x_{0}x_{−}y_{0}^{k}y_{+}^{ℓ}y_{−}^{ℓ−1});
β(x_{+}x_{−}y_{0}^{k}y_{+}^{ℓ}y_{−}^{ℓ}) = 0.

Thus, without loss of generality, we may assume G = H = 0.

Now we compute

β (∑_{j=0}^{i−1} a_{j}x_{0}x_{+}y_{0}^{2j+1}y_{+}^{i−j−1}y_{−}^{i−j})

= ∑_{j=0}^{i−1} a_{j}x_{0}x_{−}x_{+}((2j − 2i)y_{0}^{2j+2}y_{+}^{i−j−1}y_{−}^{i−j−1} + (2j + 1)y_{0}^{2j}y_{+}^{i−j}y_{−}^{i−j})

= u (a_{0}y_{+}^{i}y_{−}^{i} − 2a_{i−1}y_{0}^{2i} + ∑_{j=1}^{i−1} (a_{j}(2j + 1) + a_{j−1}(2j − 2i − 2))y_{0}^{2j}y_{+}^{i−j}y_{−}^{i−j})

Setting this equal to

uκ^{i} = u ∑_{j=0}^{i} \binom{i}{j}y_{0}^{2j}y_{+}^{i−j}y_{−}^{i−j},

and comparing the y_{0}-degree 0 terms, we obtain a_{0} = 1. We claim that a_{j} = \binom{i}{j} for j ≤ i, which we show by induction.

Assume that a_{j−1} = \binom{i}{j−1} = \binom{i}{j} \frac{j}{i−j+1}. Then the coefficient of the y_{0}-degree 2j term for 1 ≤ j ≤ i − 1 is

a_{j}(2j + 1) + a_{j−1}(2j − 2i − 2) = a_{j}(2j + 1) + \binom{i}{j − 1}(2j − 2i − 2).

Now, 2j + 1 ≢ 0 (mod p), since j ≤ i = \frac{p−3}{2}, so upon setting this equal to \binom{i}{j}, we obtain

a_{j} = \frac{1}{2j + 1} (\binom{i}{j} + \binom{i}{j}\frac{j}{i − j + 1}(2i − 2j + 2))
= \frac{1}{2j + 1} (\binom{i}{j} + 2j\binom{i}{j}) = \binom{i}{j}.

Finally, we compare the y_{0}-degree 2i terms, and find that −2a_{i−1} = \binom{i}{i} = 1. However since we also have a_{j} = \binom{i}{j}, we get −2a_{i−1} = −2\binom{i}{i−1} = −2i = 3 − p ≡ 3 ≢ 1 (mod p), and this contradiction completes the proof.

We now show that for every odd prime p, the behaviour of E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] is the same as that of E_{1}^{∗,∗}[𝔰𝔩_{2}(ℂ)] for Hodge degrees less than p − 1.

Proposition 14 (Low Hodge degree computation) For every odd prime p, E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] ≅ Λ^{∗}(u) ⊗ 𝔽_{p}[κ] for Hodge degrees less than p − 1. (The isomorphism breaks down at Hodge degree p−1, for example we have seen κ^{\frac{p−1}{2}} u = 0).

Proof We have seen that the H_{DR}^{0} = E_{1}^{∗,0}-line is generated by κ, s_{0}, s_{+}, s_{−} and so for Hodge degree less than p, the only elements are κ^{i}, 0 ≤ i ≤ \frac{p−1}{2}. We have seen that the H_{DR}^{3}-line is generated over H_{DR}^{0} by τ and u and so for Hodge degree less than p − 1, the only elements are uκ^{i}, 0 ≤ i < \frac{p−1}{2}.

Let a_{i} = dim(E_{1}^{i,0}), b_{i} = dim(E_{1}^{i,1}), c_{i} = dim(E_{1}^{i,2}), d_{i} = dim(E_{1}^{i,3}) then we have for 0 ≤ i ≤ \frac{p−1}{2}, a_{i} = d_{i} = 1 if i even and a_{i} = d_{i} = 0 for i odd. Thus in this same range, the Euler-Poincare identity gives 0 = a_{i}−b_{i}+c_{i}−d_{i} = −b_{i}+c_{i} and so b_{i} = c_{i} for 0 ≤ i ≤ \frac{p−1}{2}.

Note since E_{1}^{1,0} = 0, we know that any nonzero element of E_{1}^{0,1} would survive to E_{∞} contradicting that E_{∞} is the cohomology of a point. Thus E_{1}^{0,1} = 0 and so b_{0} = c_{0} = 0.

Now d_{1}(u) = d_{1}(x_{0}x_{−}x_{+}) = [y_{0}x_{−}x_{+}−x_{0}y_{−}x_{+}+x_{0}x_{−}y_{+}] = 0 as y_{0}x_{−}x_{+}− x_{0}y_{−}x_{+} + x_{0}x_{−}y_{+} = β(x_{0}y_{0} + \frac{1}{2}(x_{−}y_{+} + x_{+}y_{−})). Note as d_{1}(x_{0}y_{0} + \frac{1}{2}(x_{−}y_{+} + x_{+}y_{−})) = y_{0}^{2} + y_{−}y_{+}, we see directly that d_{2}(u) = κ.

This establishes then that d_{1}(uκ^{i}) = 0 and d_{2}(uκ^{i}) = [κ^{i+1}] in E_{2}. Since by proposition 13, uκ^{i} ≠ 0 in E_{1}^{2i,3} for 0 ≤ i < \frac{p−1}{2}, this forces d_{2} to be nontrivial in this range, as these elements must support some differential by E_{∞} and d_{2} is the last possible nonzero differential. Note this means also that κ^{i+1} is not hit by any d_{1}-differential for this range of i, as it must survive until E_{2} to be hit by d_{2}(uκ^{i}).

Since this accounts for the behaviour of all of the exterior 0 and 3 line in the range of Hodge degree less than p − 1, we conclude that we must have d_{1} : E_{1}^{i,2} → E_{1}^{i+1,1} is an isomorphism in the range 0 ≤ i < p−2. Thus c_{i} = b_{i+1} in this range.

Thus 0 = b_{0} = c_{0} = b_{1} = c_{1} = b_{2} = c_{2} = ⋯ = b_{p−2} = c_{p−2} which completes the proof of the proposition.

This completes the bulk of the computations. In the next section we lay out a picture of generators of E_{1}^{∗,∗} = H^{∗}(𝔰𝔩_{2}, P oly(ad^{∗})) with tables and spectral sequence diagrams to help summarize the picture. Results about the corresponding p-groups are then found.

9 Computation of E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})]

The table below gives a list of important elements in E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})]. All elements are d_{0} = β-cycles and so represent elements in E_{1}. We have seen that the elements κ, u, τ, s_{0}, s_{−}, s_{+} generate the exterior 0 and 3 lines as well as all elements of Hodge degree less than p − 1. In the table, s is the Hodge degree, t is the exterior degree and w is the weight.

s  t w
u  0  3 0 x_{0}x_{−}x_{+}
κ  2  0 0 y_{0}^{2} + y_{+}y_{−}
λ_{0} p − 1 1 0 x_{0}κ^{(p−1)/2} − x_{+}y_{0}y_{+}^{−1} (κ^{(p−1)/2} − y_{0}^{p−1})
λ_{+} p − 1 1 2p x_{+}y_{+}^{p−1}
λ_{−} p − 1 1 −2p x_{−}y_{−}^{p−1}
μ_{0} p − 1 2 0 x_{0}y_{0}^{p−2}β(y_{0}) = x_{0}y_{0}^{p−2}(x_{+}y_{−} − x_{−}y_{+})
μ_{+} p − 1 2 2p x_{+}y_{+}^{p−2}β(y_{+}) = −2x_{0}x_{+}y_{+}^{p−1}
μ_{−} p − 1 2 −2p x_{−}y_{−}^{p−2}β(y_{−}) = 2x_{0}x_{−}y_{−}^{p−1}
τ p − 1 3 0 x_{0}x_{−}x_{+}y_{0}^{p−1}
s_{0}  p  0 0 y_{0}^{p} = d_{1}(λ_{0})
s_{+}  p  0 2p y_{+}^{p} = d_{1}(λ_{+})
s_{−}  p  0 −2p y_{−}^{p} = d_{1}(λ_{−})
f_{0}  p  1 0 y_{0}^{p−1}β(y_{0}) = y_{0}^{p−1}(x_{+}y_{−} − x_{−}y_{+}) = d_{1}(μ_{0})
f_{+}  p  1 2p y_{+}^{p−1}β(y_{+}) = −2y_{+}^{p−1}(x_{+}y_{0} − x_{0}y_{+}) = d_{1}(μ_{+})
f_{−}  p  1 −2p y_{−}^{p−1}β(y_{−}) = 2y_{−}^{p−1}(x_{−}y_{0} − x_{0}y_{−}) = d_{1}(μ_{−})
γ  p  1 0 x_{+}y_{+}^{−1} (κ^{(p+1)/2} − y_{0}^{p+1}) + x_{0}y_{0}^{p}
ϵ  p  2 0 y_{0}^{p−1}(x_{−}x_{+}y_{0} − x_{0}x_{+}y_{−} + x_{0}x_{−}y_{+}) = d_{1}(τ)

Note that multiplication by y_{+}^{−1} is defined where it appears, since every term it is multiplied by contains a factor of y_{+}.

From the table above, we see the introduction of some new variables μ_{i}, λ_{i}, f_{i}, ϵ, γ. Let us explain why these give the full picture through Hodge degree p.

First note that at the location E_{1}^{p−1,1}, no differentials can be incoming from the left. Since all elements must eventually die, we see that d_{1} : E_{1}^{p−1,1} → E_{1}^{p,0} is an isomorphism. Since E_{1}^{p,0} is spanned by s_{0}, s_{−}, s_{+} by proposition 11, we see that E_{1}^{p−1,1} is spanned by elements λ_{0}, λ_{−}, λ_{+} in E_{1} such that d_{1}(λ_{i}) = s_{i}. Formulas for these elements are listed in the table above.

Now note as dim(E_{1}^{p−1,0}) = 1 = dim(E_{1}^{p−1,3}) generated by κ^{\frac{p−1}{2}} and τ respectively, by the Poincare-Euler count, we find that

dim(E_{1}^{p−1,2}) = dim(E_{1}^{p−1,1}) = 3.

Let μ_{0}, μ_{−}, μ_{+} be a basis of E_{1}^{p−1,2}. (Candidates are listed in the table above but we still need to show these candidates are linearly independent in E_{1}. This will be done in Lemma 5 below.)

Note the explicit element γ in the table has d_{1}(γ) = κ^{\frac{p+1}{2}} and so we see explicitly that [κ^{\frac{p+1}{2}} ] = 0 in E_{2}. This now leaves τ in a dilemma: it can support no nonzero d_{2} differential as the corresponding target location is zero (spanned by [κ^{\frac{p+1}{2}} ] = 0.). Thus d_{1}(τ) = ϵ ≠ 0 in E_{1}.

From what we have so far, the following short exact sequence is forced:

0 → E_{1}^{p−1,2} → E_{1}^{p,1} → E_{1}^{p+1,0} → 0.

We know that the left hand group has basis the μ_{i} and dimension 3. We also know the right hand group has basis κ^{\frac{p+1}{2}} and dimension 1. Thus E_{1}^{p,1} has dimension exactly four generated by the element γ and the d_{1}-images of the μ_{i}. In the table above we have denoted these d_{1}-images as f_{i} i.e., f_{i} = d_{1}(μ_{i}).

Now note that E_{1}^{p,3} = 0 as the exterior 3 line is generated by τ and u over E_{1}^{∗,0} and u’s contribution lies entirely before Hodge degree p − 1, while nothing nonzero times τ lies in that location. Thus we know for certain that the dimensions of E_{1}^{p,0}, E_{1}^{p,1} and E_{1}^{p,3} are 3, 4 and 0 respectively. The Euler-Poincare count now forces dim(E_{1}^{p,2}) = 1. Since we have already argued for the existence of the nonzero element ϵ in that location, it must be a basis for E_{1}^{p,2}.

We now prove that the μ_{i} and hence the f_{i} listed in the table are indeed the basis for E_{1}^{p−1,2} and E_{1}^{p,1} respectively:

Lemma 5 The elements {μ_{0}, μ_{+}, μ_{−}} in the table above are a basis for E_{1}^{p−1,2} and the elements {f_{0}, f_{+}, f_{−}} are a basis for E_{1}^{p,1}.

Proof Since the elements have distinct integral weight, to show they are linearly independent, it is sufficient to show they are nonzero. Since we also have argued that dim E_{1}^{p−1,2} = dim E_{1}^{p,1} = 3 this is enough to show they are a basis and complete the lemma.

Furthermore since d_{1}(μ_{i}) = f_{i} for i = 0, −, +, we can show either μ_{i} or f_{i} is nonzero, and it will follow that the other in the pair is also nonzero.

Recall the root lattice of 𝔰𝔩_{2} is 2ℤ, it is easy to check that E_{0}^{p−1,1}[2p], the weight 2p component of E_{0}^{p−1,1}, is one dimensional, spanned by λ_{+} = x_{+}y_{+}^{p−1}. A direct check shows that d_{0}(x_{+}y_{+}^{p−1}) = 0 and so we find that d_{0} : E_{0}^{p−1,1}[2p] → E_{0}^{p−1,2}[2p] is identically zero. Since μ_{+} ∈ E_{0}^{p−1,2}[2p] and the spectral sequence preserves weight, this shows μ_{+} is not a d_{0}-coboundary and so μ_{+} ≠ 0 ∈ E_{1}^{p−1,2} as desired. A similar proof works to show μ_{−} ≠ 0 ∈ E_{1}^{p−1,2}.

It remains to show μ_{0} ≠ 0 ∈ E_{1}^{p−1,2}. We will do this by showing d_{1}(μ_{0}) = f_{0} ≠ 0 ∈ E_{1}^{p,1}. Since f_{0} is represented by y_{0}^{p−1}β(y_{0}) and has integral weight 0, it is enough to show that y_{0}^{p−1}β(y_{0}) is not in the image of d_{0} : E_{1}^{p,0}[0] →

E_{1}^{p,1}[0]. The typical element α of E_{1}^{p,0}[0] is of the form α = ∑ℓ=0^{\frac{p−1}{2}} c_{ℓ}κ^{ℓ}y_{0}^{p−2ℓ}. Computing we find:

d_{0}(α) = (∑_{ℓ=0}^{\frac{p−1}{2}} c_{ℓ}κ_{ℓ}(p − 2ℓ)y_{0}^{p−1−2ℓ})β(y_{0}).

Now β(y_{0}) = y_{−}x_{+}−y_{+}x_{−} has the easily verified property that for any nonzero polynomial f ∈ 𝔽_{p}[κ, y_{0}], β(y_{0})f ≠ 0 ∈ E_{0}^{∗,∗}. Thus if d_{0}(α) = y_{0}^{p−1}β(y_{0}) we would have

(∑_{ℓ=0}^{\frac{p−1}{2}} c_{ℓ}(p − 2ℓ)κ_{ℓ}y_{0}^{p−1−2ℓ} − y_{0}^{p−1}) = 0.

Since y_{0} and κ are algebraically independent, equating coefficients of y_{0}^{p−1} we get c_{0}p − 1 = 0, i.e., −1 ≡ 0 mod p which is ridiculous. Thus f_{0} = y_{0}^{p−1}β(y_{0}) is not in the image of d_{0} and hence f_{0} ≠ 0 ∈ E_{1}^{p,1} and we are done.

Thus after lots of work we have found:

Theorem 18 (H^{∗}(𝔰𝔩_{2}, P oly(ad^{∗})) description). The 17 elements listed in the table above generate all parts of E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] = H^{∗}(𝔰𝔩_{2}, P oly(ad^{∗})) in Hodge degrees p or less, and on the exterior 0 and 3 lines. They are a minimal set of generators that do so. At most a finite number of additional generators lying on exterior 1, 2 lines of Hodge degree > p are required to generate the whole algebra.

Furthermore,

E_{2}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] = Λ^{∗}(u) ⊗ 𝔽_{p}[κ]/ < uκ^{\frac{p−1}{2}} , κ^{\frac{p+1}{2}} > .

Proof Most of the theorem has been proved already; we need only make the following additional observations. As E_{0} is finitely generated as a module over the Noetherian polynomial algebra R = 𝔽_{p}[s_{0}, s_{−}, s_{+}] and d_{0} is a R-module homomorphism, we conclude ker(d_{0}) and hence E_{1} are finitely generated Rmodules. It follows that E_{1} is finitely generated as an algebra and that at most a finite number of additional generators might be required.

Regardless, since we know [uκ^{\frac{p−1}{2}} ] = [κ^{\frac{p+1}{2}} ] = [s_{0}] = [s_{−}] = [s_{+}] = 0 in E_{2}, we know that no d_{2} differential is supported on the exterior 3 line, in Hodge degree p − 2 or more as E_{2}^{s,0} = 0 for s ≥ p. From this it is an easy argument to conclude that the only things that survive to the E_{2} page are as described in the theorem. (E_{∞} is known to be the cohomology of a point and d_{2} is the last possible nonzero differential.)

As a byproduct of all this analysis we have also shown:

Corollary 4 Let 𝔰𝔩_{2} denote 𝔰𝔩_{2}(𝔽_{p}) for p an odd prime and let S^{i} denote the module of homogeneous, degree i polynomials equipped with the dual adjoint action. Then for i ≥ p − 2, the following sequence is exact:

0 → H^{3}(𝔰𝔩_{2}, S^{i}) → H^{2}(𝔰𝔩_{2}, S^{i+1}) → H^{1}(𝔰𝔩_{2}, S^{i+2}) → H^{0}(𝔰𝔩_{2}, S^{i+3}) → 0.

Proof The above sequence is just

0 → E_{1}^{i,3} ⟶^{d_{1}} E_{1}^{i+1,2} ⟶^{d_{1}} E_{1}^{i+2,1} ⟶^{d_{1}} E_{1}^{i+3,0} → 0

in the spectral sequence E_{r}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})].

It is exact as E_{∞}^{s,t} = E_{2}^{s,t} = 0 for s + t > p.

Note that d_{1}(H_{DR}^{3})) = d_{1}(H_{DR}^{0}τ + H_{DR}^{0}u) = H_{DR}^{0}ϵ so any missing “ghost generator” on the exterior 2 line has to inject under d_{1} to a missing “ghost generator” on the exterior 1 line as the image of d_{1} from the exterior 3 line is accounted for.

The following diagram illustrates E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] through Hodge degree p + 1 for a typical odd prime p. The given elements are cohomology generators in Hodge degree s and exterior degree t. The dashed arrows represent the action of the differential d_{2}, and the solid arrows represent the action of the differential d_{1}.

The next diagram illustrates E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{5})] through Hodge degree 20. The numbers indicate cohomology dimensions in Hodge degree s and exterior degree t. The dashed arrows represent the differential d_{2}, which is an isomorphism in each case. The solid arrows represent the differential d_{1}, which gives a short exact sequence along each diagonal line.

Computer calculations indicate that the 17 generators we have listed are sufficient to generate the whole algebra through high Hodge degree so we conjecture:

Conjecture 1 The 17 elements u, κ, τ, ϵ, γ, λ_{i}, μ_{i}, f_{i}, s_{i} generate the algebra H^{∗}(𝔰𝔩_{2}(𝔽_{p}), P oly(ad^{∗})).

10 Bounds on the exceptional torsion in congruence subgroups

The p-group G(𝔰𝔩_{2}(𝔽_{p})) is the kernel of the mod p reduction SL_{2}(ℤ/p^{3}ℤ) → SL_{2}(𝔽_{p}). Since G(𝔤) × ℤ/pℤ (where 𝔤 is the nonabelian Lie algebra of dimension 2) is an index p subgroup of G(𝔰𝔩_{2}(𝔽_{p})), it follows that e(G(𝔰𝔩_{2}(𝔽_{p}))) ≤ pe(G(𝔤)) = p^{3}. In [Pk], it was shown that e_{∞}(G(𝔰𝔩_{2}(𝔽_{p}))) = p^{2}.

By the results in [BP], it is known that B_{2}^{∗}(G(𝔰𝔩_{2}(𝔽_{p}))) = E_{1}^{∗,∗}[𝔰𝔩_{2}(𝔽_{p})] which was computed (partially) in the last section. Using comparisons to the cyclic group of order p^{2} defined by E = F = 0, it was shown in [BP] that β_{2}(u) = β_{2}(κ) = 0 and β_{3}(u) is a nonzero multiple of κ. Thus in particular, as has been the case in all other computations, we have (E_{r}^{∗,∗}, d_{r}) = (B_{r+1}^{∗}, β_{r+1}) for 𝔰𝔩_{2}(𝔽_{p}) at least in the range between Hodge degree 0 and p − 2.

Since we have shown that uκ^{\frac{p−3}{2}} ≠ 0 in E_{1} = B_{2}^{∗}, it follows that we have argued for a nonzero differential β_{3}(uκ^{\frac{p−3}{2}} ) = cκ^{\frac{p−1}{2}} with c ≠ 0. Since it is known from [Pk] that B_{3}^{∗} is finite dimensional, it follows that κ^{\frac{p−1}{2}} lifts to an exceptional integral cohomology class of order p^{3} in H^{2p−2}(G(𝔰𝔩_{2}(𝔽_{p})), ℤ).

Thus ED(G(𝔰𝔩_{2}(𝔽_{p}))) ≥ 2p − 2.

If Conjecture 1 holds and furthermore E_{r}^{∗,∗} = B_{r+1}^{∗} in all Hodge degrees, then the inequality above would be an equality. (We feel this is almost certainly true but are unable to prove it at this time.)

11 Comments on Modular Forms and Orthogonal Steenrod Structures

11.1 Modular Forms

Following [FTY], H^{∗}(SL_{2}(ℤ), P oly_{ℂ}(V)) where V is the canonical complex representation of SL_{2}(ℤ) can be identified directly via an “Eichler-Shimura” correspondence with modular forms of certain flavors.

It is well known that the Hodge decomposition P oly_{ℂ}(V) = ⊕_{i=0}^{∞}Sym^{i}(V) yields all the finite dimensional complex irreducible representations of SL_{2}(ℂ) and equivalently 𝔰𝔩_{2}(ℂ). In this picture V = Sym^{1}(V) and ad = Sym^{2}(V). Whitehead’s lemma shows that H^{0}(SL_{2}(ℂ), P oly(V)) = H^{0}(𝔰𝔩_{2}(ℂ), P oly(V)) is the cohomology of a point and hence uninteresting. Thus it is crucial in the Eichler-Shimura correspondence that one has passed to the arithmetic subgroup SL_{2}(ℤ).

The closest analog in our work would be E_{1}^{∗,∗}[𝔰𝔩_{2}(ℤ)] = H^{∗}(𝔰𝔩_{2}(ℤ), P oly(ad^{∗})), which we have shown has p-torsion for all primes p. Here there is no longer a nice exponential correspondence between 𝔰𝔩_{2}(ℤ) and SL_{2}(ℤ) and the adjoint representation has been used, rather than the canonical representation, so there is no direct relation to modular forms, though it does seem reasonable to suspect an indirect one.

However, this arithmetic object shares properties with H^{∗}(SL_{2}(ℤ), P oly(W)), where W is the canonical integral representation of SL_{2}(ℤ), which has also been shown to have p-torsion for all primes p in unpublished work of F. Cohen, M. Salvetti and F. Callegaro.

11.2 Orthogonal Steenrod Structures.

The spectral sequence E_{r}^{∗,∗} used in this paper was motivated by the Bockstein spectral sequence of p-groups associated to p-adic Lie algebras.

For a Lie algebra 𝔤, the spectral sequence E_{r}^{∗,∗}[𝔤] arises as that of a double complex with differentials d_{0} and d_{1}.

Over 𝔽_{p}, E_{0}^{∗,∗} = Λ^{∗}(V) ⊗ P oly(V) and d_{0} is the Bockstein arising from the p-group G(𝔰𝔩_{2}(𝔽_{p})) while d_{1} is the Bockstein arising from the elementary abelian p-group. The higher Steenrod P -power operations are axiomatically determined and agree for the two p-groups.

Thus while the two p-groups have the same 𝔽_{p}-cohomology and Steenrod p-power operations, their Bocksteins act “orthogonally” in the following sense:

Definition 4 (Orthogonal Steenrod Structures) Let 𝔘 be the category of unstable modules over the mod p Steenrod algebra A_{p}. A bigraded module F^{m,n} is an orthogonal Steenrod bimodule if

(1) There are two actions of A_{p} on F^{m,n} such that the P -power operations agree under the two actions but the Bockstein operators act differently say via β_{0} and β_{1} respectively.

(2) β_{0} raises m-degree by one and preserves n-degree while β_{1} raises n-degree by one and preserves m-degree.

(3) β_{0} ∘ β_{1} = −β_{1} ∘ β_{0}.

(4) The total complex determined by the pair of commuting differentials β_{0} and β_{1} is acyclic.

(Note the bigrading above does not coincide with the one we have been using in the paper, a regrading was made for convenience and is explained in the appendix.)

In our case the resulting spectral sequence derived from these anti-commuting Bocksteins proved very useful. Considering that every indecomposable injective 𝔘-module is of the form L⊗J[n] where L is a summand of the cohomology of an elementary abelian p-group and J[n] is a Brown-Gitler module (see [Sc]), it seems that considering which equivalence classes of extensions in Ext_{𝔘}(k, k) can be equipped with such “orthogonal structures” might be interesting.

12 Acknowledgments

We would like to thank Fred Cohen and John Harper for many interesting and useful discussions regarding the topics in this paper.

A Construction of the Algebraic Spectral Sequence E[𝔤]

In this section, we will derive a spectral sequence for Lie algebra cohomology which is based on the Bockstein spectral sequence of the groups mentioned in the introduction but is defined for Lie algebras over an arbitrary PID k. We give what are probably too many details so the reader is referred to the short summary of the final results given in an earlier section.

Let k be an arbitrary PID. Let 𝔏 be a k-Lie algebra and V be a finite dimensional 𝔏-module. Set V^{(s)} = V ⊗ ⋯ ⊗ V (s times) for s ≥ 1 and V^{(0)} = k. V^{(s)} is given the usual 𝔏-module structure, i.e.

\begin{matrix}u · (v_{1} ⊗ ⋯ ⊗ v_{s}) = & (u · v_{1} ⊗ ⋯ ⊗ v_{s}) \\ + (v_{1} ⊗ u · v_{2} ⊗ ⋯ ⊗ v_{s}) + ⋯ + (v_{1} ⊗ ⋯ ⊗ u · v_{s}) \end{matrix}

for all u ∈ 𝔏. (V^{(0)} is given the trivial action). The 𝔏-action on V is extended to V^{(s)} by saying it acts via derivations.

Let T^{∗}(V) = k ⊕ V ⊕ V^{(2)} ⊕ … be the usual tensor algebra. We will make it a graded algebra by setting V^{(n)} to have grading 2n. Thus T^{∗}(V) becomes a 𝔏-module by the actions defined above. Let I be the ideal generated by elements v ⊗ w − w ⊗ v for all v, w ∈ V. Then it is easy to check that the action of 𝔏 preserves I. Thus S^{∗}(V) = T^{∗}(V)/I is a graded algebra which inherits an 𝔏 action from T^{∗}(V). Note that the 𝔏 action preserves the grading of S^{∗}(V) so that S^{i}(V) is a 𝔏-module for all i ≥ 0. It is well known that S^{∗}(V) is a polynomial algebra on k variables where k = dim(V). Fix u ∈ 𝔏; then for v_{1}, …, v_{s} ∈ V we have

u · (v_{1} … v_{s}) = ∑_{i=1}^{s} v_{1} … v_{i−1} · (u · v_{i}) · v_{i+1} … v_{s},

so u acts as a derivation on S^{∗}(V). Given a 𝔏-module M let ∧^{∗}(𝔏, M) be the usual Koszul resolution for H^{∗}(𝔏; M). Thus ∧^{i}(𝔏, M) consists of M-valued alternating i-forms on 𝔏 and

\begin{aligned}(dω)(x_{0}, …, x_{s}) = & ∑_{i<j}(−1)^{i+j}ω([x_{i}, x_{j}], x_{0}, … x̂_{i}, …, x̂_{j}, …, x_{s}) \\ & + ∑_{i=0}^{s}(−1)^{i}x_{i} · ω(x_{0}, …, x̂_{i}, …, x_{s}) \end{aligned}

for all ω ∈ ∧^{s}(𝔏, M) and all x_{0}, …, x_{s} ∈ 𝔏.

Consider

H = ⨁_{i=0}^{∞} ∧^{∗}(𝔏, S^{i}(V)),

the direct sum of Koszul complexes equipped with differential d the direct sum of the differentials of each of the complexes involved. Using the algebra structure of S^{∗}(V) one can give H an algebra structure with grading where the grading of ∧^{1}(𝔏, S^{0}(V)) is 1 and that of ∧^{0}(𝔏, S^{1}(V)) is 2. With these conventions H is isomorphic to ∧^{∗}(x_{1}, …, x_{n}) ⊗ k[s_{1}, …, s_{k}] where n = dim(𝔏), x_{1}, …, x_{n} a dual basis to a basis of 𝔏, k = dim(V) and s_{1}, …, s_{k} a basis of V. We will show that the differential d on H is a derivation with respect to this algebra structure. Thus

d(uv) = (du)v + (−1)^{deg(u)}u(dv)

for u, v homogeneous elements in H. First note that d is a derivation on ∧^{∗}(𝔏, S^{0}(V)). This is because this is the same as d for ∧^{∗}(𝔏, k) which is well known to be a derivation. Also on ∧^{0}(𝔏, S^{∗}(V)) = S^{∗}(V) we have

d(s_{i_{1}} … s_{i_{m}})(u) = u · (s_{i_{1}} … s_{i_{m}})
= ∑_{l=1}^{m} s_{i_{1}} … u · s_{i_{l}} … s_{i_{m}}

= [∑_{l=1}^{m} s_{i_{1}} … (ds_{i_{l}}) … s_{i_{m}}] (u)

for all u ∈ 𝔏. Thus d is a derivation on ∧^{0}(𝔏, S^{∗}(V)). Now the elements

x_{μ_{1}} … x_{μ_{t}}s_{λ_{1}} … s_{λ_{m}}

for μ_{1} < ⋯ < μ_{t} and λ_{i} arbitrary form a basis for H. Let xμ̄ = x_{μ_{1}} … x_{μ_{t}} and sλ̄ = s_{λ_{1}} … s_{λ_{m}}. We also say l(λ̄) = m etc. So using this sort of notation let xμ̄sλ̄ and xη̄sκ̄ be two arbitrary basis elements. Then

d(xμ̄sλ̄xη̄sκ̄) = d(xμ̄xη̄sλ̄sκ̄).

Suppose we have the identity

d(xμ̄sλ̄) =d(xμ̄)sλ̄ + (−1)^{l(μ̄)}xμ̄d(sλ̄),  (4)

then we have

d(xμ̄sλ̄xη̄sκ̄) = d(xμ̄xη̄)sλ̄sκ̄ + (−1)^{l(μ̄)+l(η̄)}xμ̄xη̄d(sλ̄sκ̄)
= d(xμ̄)xη̄sλ̄sκ̄ + (−1)^{l(μ̄)}xμ̄d(xη̄)sλ̄sκ̄
+(−1)^{l(μ̄)+l(η̄)}xμ̄xη̄[d(sλ̄)sκ̄ + sλ̄d(sκ̄)].

On the other hand,

d(xμ̄sλ̄)xη̄sκ̄ + (−1)^{l(μ̄)}xμ̄sλ̄d(xη̄sκ̄)
= d(xμ̄)xη̄sλ̄sκ̄ + (−1)^{l(μ̄)+l(η̄)}xμ̄xη̄sκ̄d(sλ̄) + (−1)^{l(μ̄)}xμ̄d(xη̄)sλ̄sκ̄
+ (−1)^{l(μ̄)+l(η̄)}xμ̄xη̄sλ̄d(sκ̄).

Comparing the two expressions we get

d(xμ̄sλ̄xη̄sκ̄) = d(xμ̄sλ̄)xη̄sκ̄ + (−1)^{l(μ̄)}xμ̄sλ̄d(xη̄sκ̄).

So using an easy linearity argument we see d is a derivation on H. So it remains to prove equation 4. To do this let {e_{1}, … e_{n}} be a basis of 𝔏 with x_{i}(e_{j}) = δ_{i,j} the Kronecker delta function. Then for α_{0} < ⋯ < α_{t} we have

d(xμ̄sλ̄)(e_{α_{0}}, …, e_{α_{t}}) = ∑_{i<j}(−1)^{i+j}(xμ̄sλ̄)([e_{α_{i}}, e_{α_{j}}], e_{α_{0}}, .., ê_{α_{i}}, .., ê_{α_{j}} , .., e_{α_{t}})

+ ∑_{i=0}^{t}(−1)^{i}[xμ̄e_{α_{i}} · sλ̄](e_{α_{0}}, …, ê_{α_{i}}, …, e_{α_{t}})

= d(xμ̄)s_{λ}(e_{α_{0}}, …, e_{α_{t}})

+ ∑_{i=0}^{t}(−1)^{i}[xμ̄e_{α_{i}} · sλ̄](e_{α_{0}}, …, ê_{α_{i}}, …, e_{α_{t}}).

So to show equation 4, it is enough to show

∑_{i=0}^{t}(−1)^{i}[xμ̄e_{α_{i}} · sλ̄](e_{α_{0}}, …, ê_{α_{i}}, …, e_{α_{t}}) = (−1)^{t}xμ̄(dsλ̄)(e_{α_{0}}, …, e_{α_{t}}).

This is done by an easy case-by-case analysis and is left to the reader.

A.1 Differentials

In the last section we showed that given a finite dimensional 𝔏-module V, the differential d on H = ⊕_{i=0}^{∞} ∧^{∗} (𝔏, S^{i}(V)) given by the direct sum of the Koszul differentials is a derivation. We now discuss a method of putting differentials D : H → H on H. This means D is linear, D ∘ D = 0 and D is a derivation with respect to the graded algebra structure of H. We choose a basis {e_{1}, …, e_{n}} of 𝔏 and let {x_{1}, …, x_{n}} be a dual basis. Let {s_{1}, …, s_{k}} be a basis of V. Then, as discussed before,

H = ∧^{∗}(x_{1}, …, x_{n}) ⊗ k[s_{1}, …, s_{k}]

as graded algebras where deg(x_{i}) = 1, deg(s_{i}) = 2. We will use this notation freely in the proof of the next lemma.

Lemma 6 Let H be as in the paragraph above. Then if

ψ : ∧^{1}(𝔏, S^{0}(V)) → ∧^{0}(𝔏, S^{1}(V))

is a linear map then it extends to a unique derivation

D : H → H

which vanishes on

∧^{0}(𝔏, S^{∗}(V)) = S^{∗}(V).

Furthermore the derivation is a differential i.e. D ∘ D = 0.

Proof First let us define D as a linear map H → H. For α_{0} < ⋯ < α_{s} define

D(x_{α_{0}} … x_{α_{s}}) = ∑_{l=0}^{s}(−1)^{l}x_{α_{0}} … ψ(x_{α_{l}}) … x_{α_{s}}

and

D(1) = 0.

As {x_{α_{0}} … x_{α_{s}} : −1 ≤ s < n, α_{0} < ⋯ < α_{s}} is a basis for ∧^{∗}(𝔏, S^{0}(V)) we see we have defined a unique linear map

D : ∧^{∗}(𝔏, S^{0}(V)) → H

extending ψ. It is routine to show (and is left to the reader) that D defined in this way is a derivation ∧^{∗}(𝔏, S^{0}(V)) → H. Now we extend D to ∧^{0}(𝔏, S^{∗}(V)) = S^{∗}(V) by defining it to be identically 0 on this part. Finally we define D on an arbitrary basis element xᾱsλ̄ of H by

D(xᾱsλ̄) = D(xᾱ)sλ̄.

Note this agrees with the parts already defined and hence defines a linear map D : H → H. It is now routine (and left to the reader) to verify that D is a derivation with respect to the graded algebra structure on H. Now we argue why this is the unique extension of ψ to such a derivation which vanishes on ∧^{0}(𝔏, S^{∗}(V)). Clearly the definition of D on ∧^{∗}(𝔏, S^{0}(V)) was forced if we want a derivation. Then as we require D to vanish on ∧^{0}(𝔏, S^{∗}(V)) we see that the definition on an arbitrary basis element is also forced, and uniqueness follows. Note that D raises grading by 1. Now we are left only to show D ∘ D = 0 on H. As D is linear it is enough to check on basis elements xᾱsλ̄ of H. We have

D ∘ D(xᾱsλ̄) = D(D(xᾱ)sλ̄)
= D(D(xᾱ))sλ̄ + (−1)^{l(ᾱ)+1}D(xᾱ)D(sλ̄)
= D(D(xᾱ))sλ̄.

So we will be done once we show D(D(xᾱ)) = 0. For this let ᾱ be α_{1} < ⋯ < α_{s} for some 1 ≤ s ≤ n. Note for s = 1 we have D(x_{α_{1}}) ∈ ∧^{0}(𝔏, S^{1}(V)) so we have D(D(x_{α_{1}})) = 0. Also D(D(1)) = 0. So we will prove D(D(xᾱ)) = 0 by induction on s = l(ᾱ). Assume we have shown it for s < l where l > 1; then

D(D(x_{α_{1}} … x_{α_{l}})) = D(D(x_{α_{1}})x_{α_{2}} … x_{α_{l}} − x_{α_{1}}D(x_{α_{2}} … x_{α_{l}}))
= D(D(x_{α_{1}}))x_{α_{2}} … x_{α_{l}} + D(x_{α_{1}})D(x_{α_{2}} … x_{α_{l}})
−D(x_{α_{1}})D(x_{α_{2}} … x_{α_{l}}) + x_{α_{1}}D(D(x_{α_{2}} … x_{α_{l}}))
= 0,

where in the last step we used cancellation and the inductive hypothesis. Thus by induction we are done and we have shown D ∘ D = 0 on H, as desired.

Now we study when the differential D constructed in Lemma 6 has the additional property d ∘ D = −D ∘ d on H, where d is the Koszul differential of H as in section 1. Let {e_{1}, …, e_{n}} be a basis for 𝔏 as before, {x_{1}, …, x_{n}} the dual basis, and c_{ij}^{k} = [e_{i}, e_{j}]_{k} the structure constants of the Lie algebra 𝔏. Let {s_{1}, …, s_{k}} be a basis of V as before.

Lemma 7 Assume the notation of the preceding paragraph. Let

ψ : ∧^{1}(𝔏, S^{0}(V)) → ∧^{0}(𝔏, S^{1}(V))

be a linear map. Let D be the differential extending ψ as given by Lemma 6. Then d ∘ D = −D ∘ d if and only if

e_{j} · ψ(x_{i}) = ∑_{l=1}^{n} c_{lj}^{i}ψ(x_{l})

and

∑_{l=1}^{n}(e_{l} · s_{t})ψ(x_{l}) = 0

for all 1 ≤ i, j ≤ n and 1 ≤ t ≤ k.

Proof Since D and d are linear it is enough to check d ∘ D = −D ∘ d on basis elements xᾱsλ̄. So we have

(d ∘ D)(xᾱsλ̄) = d(D(xᾱ)sλ̄) = d(D(xᾱ))sλ̄ + (−1)^{l(ᾱ)+1}D(xᾱ)d(sλ̄).

On the other hand we have

(−D ∘ d)(xᾱsλ̄) = −D(d(xᾱ)sλ̄ + (−1)^{l(ᾱ)}xᾱd(sλ̄))
= −D(d(xᾱ))sλ̄ + (−1)^{l(ᾱ)+1}D(xᾱ)d(sλ̄) − xᾱD(d(sλ̄)).

So we see by comparing the two expressions that we have d ∘ D = −D ∘ d if we can show d(D(xᾱ)) = −D(d(xᾱ)) and D(d(sλ̄)) = 0. Let us work with the second condition. Let sλ̄ = s_{λ_{1}} … s_{λ_{m}} for some m ≥ 1 then

D(d(sλ̄)) = D (∑_{i=1}^{m} s_{λ_{1}} … d(s_{λ_{i}}) … s_{λ_{m}})

= D (∑_{i=1}^{m} d(s_{λ_{i}})s_{λ_{1}} … ŝ_{λ_{i}} … s_{λ_{m}})

= ∑_{i=1}^{m} D(d(s_{λ_{i}}))s_{λ_{1}} … ŝ_{λ_{i}} … s_{λ_{m}}.

So we see that to show D(d(sλ̄)) = 0 one needs only show D(d(s_{i})) = 0 for all 1 ≤ i ≤ k. Now we work with the first condition above. Suppose one has shown d(D(x_{i})) = −D(d(x_{i})) for 1 ≤ i ≤ n. Then let us use induction on t = l(ᾱ) to show d(D(xᾱ)) = −D(d(xᾱ)). So by hypothesis we have this for t = 1 so assume we have shown it for t < l for some l > 1. Then one has:

−D(d(x_{α_{1}} … x_{α_{l}})) = −D(d(x_{α_{1}})x_{α_{2}} … x_{α_{l}} − x_{α_{1}}d(x_{α_{2}} … x_{α_{l}}))
= −D(d(x_{α_{1}}))x_{α_{2}} … x_{α_{l}} − d(x_{α_{1}})D(x_{α_{2}} … x_{α_{l}})
+D(x_{α_{1}})d(x_{α_{2}} … x_{α_{l}}) − x_{α_{1}}D(d(x_{α_{2}} … x_{α_{l}}))
= d(D(x_{α_{1}}))x_{α_{2}} … x_{α_{l}} − d(x_{α_{1}})D(x_{α_{2}} … x_{α_{l}})
+D(x_{α_{1}})d(x_{α_{2}} … x_{α_{l}}) + x_{α_{1}}d(D(x_{α_{2}} … x_{α_{l}}))
= d(D(x_{α_{1}})x_{α_{2}} … x_{α_{l}} − x_{α_{1}}D(x_{α_{2}} … x_{α_{l}}))
= d(D(x_{α_{1}} … x_{α_{l}})).

Thus by induction we have the second condition. We conclude from all the above that if we have d(D(x_{i})) = −D(d(x_{i})) for all 1 ≤ i ≤ n and D(d(s_{i})) = 0 for all 1 ≤ i ≤ k then we have d ∘ D = −D ∘ d on H as desired. Let us translate the condition d(D(x_{i})) = −D(d(x_{i})) noting that D(x_{i}) = ψ(x_{i}) ∈ ∧^{0}(𝔏, S^{1}(V)) = S^{1}(V). Thus d(D(x_{i}))(e_{j}) = e_{j} · ψ(x_{i}). On the other hand,

−D(d(x_{i}))(e_{j}) = D ( ∑_{l<m} c_{lm}^{i}x_{l}x_{m}) (e_{j})

= ∑_{l<m} c_{lm}^{i}[D(x_{l})x_{m} − x_{l}D(x_{m})](e_{j})

= ∑_{l<j} c_{lj}^{i}ψ(x_{l}) − ∑_{j<m} c_{jm}^{i}ψ(x_{m})

= ∑_{l<j} c_{lj}^{i}ψ(x_{l}) + ∑_{j<m} c_{mj}^{i}ψ(x_{m})

= ∑_{l=1}^{n} c_{lj}^{i}ψ(x_{l}).

So d(D(x_{i})) = −D(d(x_{i})) for all 1 ≤ i ≤ n if and only if

e_{j} · ψ(x_{i}) = ∑_{l=1}^{n} c_{lj}^{i}ψ(x_{l})

for all 1 ≤ i, j ≤ n. Now let us translate the condition D(d(s_{i})) = 0. Note d(s_{i}) ∈ ∧^{1}(𝔏, S^{1}(V)) is a 1-form on 𝔏 with values in S^{1}(V) = V so as d(s_{i})(u) = u · s_{i} for all u ∈ 𝔏 we have d(s_{i}) = ∑_{j=1}^{n} d(s_{i})(e_{j})x_{j} = ∑_{j=1}^{n}(e_{j} · s_{i})x_{j}. Thus

D(d(s_{i})) = D (∑_{j=1}^{n}(e_{j} · s_{i})x_{j} ) = ∑_{j=1}^{n}(e_{j} · s_{i})ψ(x_{j})

where we have used that D(e_{j} · s_{i}) = 0 as e_{j} · s_{i} ∈ ∧^{0}(𝔏, S^{1}(V)). Thus D(d(s_{i})) = 0 for all 1 ≤ i ≤ k if and only if ∑_{j=1}^{n}(e_{j} · s_{i})ψ(x_{j}) = 0 for all 1 ≤ i ≤ k. This ends the proof of the lemma.

A.2 The spectral sequence

Now we apply the results of the last section to obtain a spectral sequence for Lie algebra cohomology. Let 𝔏 be a Lie algebra over the PID k, and V = ad^{∗} be the 𝔏-module which is the dual space 𝔏^{∗} of 𝔏 with action

(u · α)(v) = α([v, u])

for u, v ∈ 𝔏 and α ∈ 𝔏^{∗}. It is easy to check using the Jacobi identity that this is indeed a 𝔏-module. Fix a basis {e_{1}, …, e_{n}} of 𝔏. Let {x_{1}, …, x_{n}} be the dual basis viewed in ∧^{1}(𝔏, S^{0}(V)) = ∧^{1}(𝔏, k) and {s_{1}, …, s_{n}} the dual basis viewed in ∧^{0}(𝔏, S^{1}(V)) = V. Then let ψ : ∧^{1}(𝔏, S^{0}(V)) → ∧^{0}(𝔏, S^{1}(V)) be the linear map given by taking x_{i} to s_{i} for all 1 ≤ i ≤ n. Recall H and d from the last sections. By Lemma 6 one has ψ extends to a differential D on H. Note we have

e_{j} · s_{i} = ∑_{l=1}^{n}[e_{l}, e_{j}]_{i}s_{l} = ∑_{l=1}^{n} c_{lj}^{i}s_{l},

where the first equality is verified by noting both sides evaluate to the same thing when evaluated on e_{k} for 1 ≤ k ≤ n and the second equality uses the definition of the structure constants c_{lj}^{i}. Thus as ψ(x_{i}) = s_{i} we have

e_{j} · ψ(x_{i}) = ∑_{l=1}^{n} c_{lj}^{i}ψ(x_{l}).

We also have

∑_{j=1}^{n}(e_{j} · s_{i})ψ(x_{j}) = ∑_{j=1}^{n}(e_{j} · s_{i})s_{j} = ∑_{j,l=1}^{n} c_{lj}^{i}s_{l}s_{j} = 0.

Thus we have verified the conditions of Lemma 7 so we conclude d ∘ D = −D ∘ d. Thus we have two commuting differentials D, d on H = ⊕_{i=0}^{∞} ∧^{∗} (𝔏, S^{∗}(ad^{∗})) which we note is bigraded. Note S^{i}(ad^{∗}) is isomorphic to the vector space of symmetric i-forms on 𝔏. We will use S^{i} = S^{i}(ad^{∗}) for short. Recall we can view

H = ∧^{∗}(x_{1}, …, x_{n}) ⊗ k[s_{1}, …, s_{n}]

as graded algebras where deg(x_{i}) = 1, deg(s_{i}) = 2 and when we do this D is a derivation. But D(x_{i}) = s_{i} for all 1 ≤ i ≤ n by construction so we see that the cohomology of the complex (H, D) is concentrated in the zero grading where it is k. (This follows from Künneth’s theorem for example). Now let

F_{0}^{s,q} = ∧^{q−s}(𝔏, S^{s})

for s, q ≥ 0. Then we see d gives a vertical differential F_{0}^{s,q} → F_{0}^{s,q+1} and D gives a horizontal differential F_{0}^{s,q} → F_{0}^{s+1,q} and D, d commute (up to sign). Such a complex is called a double complex. If we let M be the graded complex with M^{ℓ} = ⊕_{s+q=ℓ}F_{0}^{s,q} and differential T = D + d then one gets by a standard construction two first quadrant F_{0} spectral sequences with F_{0}^{∗,∗} given by the formula above converging to the cohomology of (M^{∗}, T). (See [B]). One of these spectral sequences has F_{1}^{∗,∗} equal to the cohomology of F_{0} with respect to D. However we have argued before that this is acyclic so F_{1}^{s,q} = 0 for s+q > 0 and F_{1}^{0,0} = k. Thus F_{∞}^{s,q} = 0 for s+q > 0 and F_{∞}^{0,0} = k for this spectral sequence. Since this abuts to the cohomology of M we conclude that (M^{∗}, T) is an acyclic complex. (This means that the only nonzero cohomology is in the zero grading and H^{0}(M^{∗}; k) = k). The other spectral sequence has F_{1}^{∗,∗} equal to the cohomology of F_{0} with respect to d. Thus it is easy to see F_{1}^{s,q} = H^{q−s}(𝔏, S^{s}) for s, q ≥ 0. However this spectral sequence also abuts to the cohomology of M^{∗} so we conclude it also has F_{∞}^{s,q} = 0 for s + q > 0 and F_{∞}^{0,0} = k. Also as D and d were derivations one concludes as usual that all the differentials d_{r} in this spectral sequence are derivations with respect to the induced graded algebra structures on the F_{r}. We summarize in the following theorem:

Theorem 19 Fix an arbitrary PID k. Let 𝔏 be a k-Lie algebra. Let S^{s} be the 𝔏-module of symmetric s-forms on 𝔏 with the dual adjoint action discussed before. Then there is a first quadrant F_{0}-spectral sequence with F_{0}^{s,q} = Λ^{q−s}(𝔏, S^{s}) and F_{1}^{s,q} = H^{q−s}(𝔏, S^{s}) for s, q ≥ 0 which has F_{∞}^{s,q} = 0 for s + q > 0 and F_{∞}^{0,0} = k. Furthermore all the differentials d_{r} : F_{r}^{s,q} → F_{r}^{s+r,q−r+1} are derivations with respect to the induced graded algebra structure on F_{r}.

In the above spectral sequence, s denotes the polynomial degree, q − s represents the exterior degree and (q − s) + 2s = q + s represents the total degree. This unusual grading is used so that in the derivation, the initial differentials would be horizontal and vertical. However, this bigrading is not so easy to work with in applications so finally we will tweak the definition of the bigrading of the above spectral to:

E_{0}^{s,t} = F_{0}^{s,s+t}.

This yields the final form of the main spectral sequence that we use in this paper:

Theorem 20 Fix an arbitrary PID k and k-Lie algebra 𝔏. Let S^{s} be the 𝔏-module of symmetric degree s-polynomials on 𝔏, equipped with the dual adjoint action.

Then there is a first quadrant spectral sequence

E_{0}^{s,t} = Λ^{t}(𝔏, S^{s})

with differentials d_{r} : E_{r}^{s,t} → E_{r}^{s+r,t−(2r−1)} which are derivations with respect to the algebra structure induced from

E_{0}^{∗,∗} = Λ^{∗}(𝔏^{∗}) ⊗ S^{∗}(𝔏^{∗}).

The number s is called the polynomial (or Hodge) degree, t the exterior degree and 2s + t the total degree. Thus note the differential d_{r} raises polynomial degree by r while raising total degree by one (and hence lowers exterior degree by (2r − 1)).

The spectral sequence converges to the cohomology of a point. Explicit formulas for the differentials are stated within the paper itself and are easily derived from the construction in the appendix.

References

[A] Adem, A: On the exponent of cohomology of discrete groups, Bull. London Math. Soc. 21, 585–590 (1989)

[AP] Adem, A., Pakianathan, J.: On the cohomology of central Frattini extensions, J. Pure Appl. Algebra 159, 1–14 (2001)

[BH] Baez J., Huerta J.: The algebra of grand unified theories, Bull. Amer. Math. Soc. 47, 483–552 (2010)

[Be] Benson, D.J.: Representations and Cohomology II. Cambridge University Press, Cambridge (1991)

[Bo] Bourbaki, N.: Lie groups and Lie algebras, Chapters I-III. Springer-Verlag, New York (1989)

[Br] Browder, W.: Cohomology and Group Actions, Invent. Math. 71, 599–607 (1983)

[BP] Browder, W., Pakianathan, J.: Cohomology of uniformly powerful p-groups, Trans. Amer. Math. Soc. 352, 2659-2688 (2000)

[B] Brown, K.S.: Cohomology of Groups. Springer Verlag, New York (1994)

[BT] Bott, R., Tu, L.: Differential Forms in Algebraic Topology. Springer Verlag, New York (1982)

[Co] Coutinho, S.C.: A Primer of Algebraic D-modules. Cambridge University Press, Cambridge (1995)

[CS] Chas, M., Sullivan, D.: Closed string operators in topology leading to Lie bialgebras and higher string algebra, Preprint, arXiv:math.GT/0212358 (2010)

[DS] Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic pro-p groups. Cambridge University Press, Cambridge (1991)

[FH] Fulton, W., Harris, J.: Representation Theory. Springer Verlag, New York (1991)

[FTY] Furusawa, M., Tezuka, M., Yagita, N.: On the cohomology of classifying spaces of torus bundles and automorphic forms, J. London Math. Soc. (2) 37, 520–534 (1988)

[GG] Goto, M., Grosshans, F.D.,: Semisimple Lie algebras. Marcel Dekker, New York [GW] Gruher, K., Westerland, C.: String topology prospectra and Hochschild cohomology,

J. Topol. 1 837–856 (2008)

[Le] Leary, I.J.: A bound on the exponent of the cohomology of BC-bundles. In: Algebraic Topology: new trends in localization and periodicity, pp. 255–259. Birkhäuser, Basel (1996)

[Ma] Malm, E.: String topology and the Hochschild cohomology of the based loop space, Preprint (2010)

[MT] Mimura, M., Toda, H.: Topology of Lie groups I, II. American Mathematical Society, Providence (1991)

[NP] Nam, K.B, Pakianathan, J.: On generalized Witt algebras in one variable, To appear, Turkish Journal of Mathematics (2011)

[Pk] Pakianathan, J.: Exponents and the cohomology of finite groups, Proc. Amer. Math. Soc. Vol. 128, 1893–1897 (2000)

[PY1] Pakianathan, J., Yalçın, E.: Quadratic maps and Bockstein closed group extensions, Trans. Amer. Math. Soc. Vol. 359 6079–6110 (2007)

[PY2] Pakianathan, J., Yalçın, E.: Bockstein closed 2-group extensions and cohomology of quadratic maps, arXiv 0075818, Preprint (2010)

[Ro] Robert, A.: A Course in p-adic Analysis. Springer Verlag, New York (2000)

[Sc] Schwartz, L.: Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture. University of Chicago Press, Chicago (1994)

[Sh] Shimura, G.: Automorphic Functions and Number Theory. Springer, Berlin (1968)