REPRESENTATIONS OF FINITE DIMENSIONAL HOPF ALGEBRAS

Martin Lorenz

Department of Mathematics Temple University Philadelphia, PA 19122-6094

                            e-mail: lorenz@math.temple.edu

Abstract. Let H denote a finite dimensional Hopf algebra with antipode S over a field 𝕜. We give a new proof of the fact, due to Oberst and Schneider [OS], that H is a symmetric algebra if and only if H is unimodular and S^{2} is inner. If H is involutory and not semisimple, then the dimensions of all projective H-modules are shown to be divisible by char 𝕜. In the case where 𝕜 is a splitting field for H, we give a formula for the rank of the Cartan matrix of H, reduced mod char 𝕜, in terms of an integral for H. Explicit computations of the Cartan matrix, the ring structure of G_{0}(H), and the structure of the principal indecomposable modules are carried out for certain specific Hopf algebras, in particular for the restricted enveloping algebras of completely solvable p-Lie algebras and of sl(2, 𝕜).

Introduction

This article is a study of representations of finite dimensional Hopf algebras H in the spirit of Larson’s “Characters of Hopf Algebras” [L], but with the emphasis on the non-semisimple case. Thus particular attention is given to the properties of projective modules. To a large extent, we work inside the Grothendieck groups G_{0}(H) and K_{0}(H) of the categories of finitely generated H-modules and finitely generated projective H-modules, respectively, and the various connections between G_{0}(H) and K_{0}(H) are among our main focal points: The comultiplication of H causes G_{0}(H) to be a ring and K_{0}(H) to be a module over G_{0}(H); there is a canonical duality between K_{0}(H) and G_{0}(H) which has a very natural interpretation in terms of Hattori-Stallings ranks (for K_{0}(H)) and ordinary characters (for G_{0}(H)); and, of course, there is the Cartan map c : K_{0}(H) → G_{0}(H).

Here are the main results of the article. Throughout, H denotes a finite dimensional Hopf algebra over a field 𝕜 of characteristic p ≥ 0, and S is the antipode of H.

Theorem 1. If H is involutory (that is, S^{2} = Id) and not semisimple, then p divides the dimension of every projective H-module.

The next result determines the rank of the map c̃ = Id_{𝕜} ⊗c : 𝕜 ⊗_{ℤ} K_{0}(H) → 𝕜 ⊗_{ℤ} G_{0}(H). This is a lower bound for the rank of c, and the two ranks are identical for p = 0. We let C denote the Cartan matrix, that is, the matrix of the Cartan map with respect to the canonical ℤ-bases of G_{0}(H) and K_{0}(H) that are afforded by the irreducible H-modules and their projective covers, respectively.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16W30, 16G99.

Key words and phrases. Finite dimensional Hopf algebra, character, Hattori-Stallings rank, Grothendieck group, Cartan matrix, projective module, principal indecomposable module, duality, trace function, orthogonality, symmetric algebra, group algebra, restricted enveloping algebra, p-Lie algebra.

The author was supported in part by NSF Grant DMS–9400643.

Theorem 2. Suppose that k is a splitting field for H. Then

rank c̃ = dim_{𝕜}(H ◁ t) .

Here t is any nonzero left integral of H and ◁ denotes the right adjoint action of H on itself. Moreover, if S^{2} is inner, then H is semisimple iff C = Id iff p does not divide det C.

The article also contains explicit computations of the Cartan matrix C and of the ring structure of G_{0}(H) for a number of specific Hopf algebras H that are of interest, in particular for the restricted enveloping algebras of completely solvable p-Lie algebras and of sl(2, 𝕜). These algebras display features that contrast sharply with the classical case of finite group algebras: C is always singular, and the ring G_{0}(H) is not semiprime for the restricted enveloping algebra of sl(2, 𝕜).

A brief summary of the contents of the individual sections is as follows.

§1 reviews the basic pertinent material on Grothendieck groups in the more general setting of finite dimensional associative algebras. This section is, to a large degree, a summary of parts of [Ba2], specialized to finite dimensional algebras.

§2, on symmetry and dimensions, is independent of Grothendieck groups and is entirely based on a few simple observations about duality and traces. Theorem 1 is proved in §2.3 and, in §2.5, unimodularity of H is shown to be equivalent with self-duality of the projective cover of the “trivial” H-module. As a consequence, one obtains that H is a symmetric algebra if and only if H is unimodular and S^{2} is inner.

§3 takes up the material of §1 in the context of Hopf algebras and contains the proof of Theorem 2 (in §3.4). The main emphasis is on the ring and module structures that are now carried by the various objects of §1. This section also contains an analysis of the special case where the Jacobson radical of H is a Hopf ideal.

§4 is devoted to a detailed discussion of some explicit examples: the Sweedler algebra H_{4}, a class of algebras that were constructed by Radford [R3] (based on earlier examples due to Taft [T]), and the restricted enveloping algebras of completely solvable p-Lie algebras and of sl(2, 𝕜). In each case, the ring G_{0}(H), its module K_{0}(H), the principal indecomposable modules, and the Cartan matrix are determined.

The author would like to thank Susan Montgomery for her thorough reading of the first version of this article which lead to a number of improvements, clarifications, and proper accreditations. In particular, she pointed out to us that the aforementioned characterization of symmetry was first observed by Oberst and Schneider [OS]. The present proof is a simplification of our original argument which incorporates a suggestion of hers and H.-J. Schneider. After the first version of the article was circulated, the author learned from Jim Humphreys that most of the features of the sl(2, 𝕜)- example exhibited in §4.4 have previously been discovered by Pollack [Po] and have been rederived by Humphreys in [H]. There is also previous work of Humphreys on symmetry [H2]. Related material on representations of finite dimensional cocommutative Hopf algebras, phrased in the language of (finite) algebraic groups, can be found in [V].

Notations and conventions. All algebras, Hopf and otherwise, considered in this article are finite dimensional over a commutative base field 𝕜 of characteristic p ≥ 0, and all modules are left modules and are assumed to be finite dimensional over 𝕜. Finally, ⊗ stands for ⊗_{𝕜} and .^{∗} = Hom_{𝕜}( . , 𝕜) denotes the linear dual. Further assumptions will be explicitly stated at the beginning of each section.

1. Background from the theory of finite dimensional algebras

Throughout this section, A will denote a (finite dimensional) algebra over 𝕜 and J = rad A is the Jacobson radical of A. We fix a full set of nonisomorphic irreducible A-modules V_{1}, …, V_{t} and we let P_{i} = P (V_{i}) denote their projective covers, the principal indecomposable A-modules (cf. [CR], p. 131).

1.1 Grothendieck groups. Let G_{0}(A) denote the Grothendieck group of the category of (finite dimensional left) A-modules. This is the abelian group that is generated by the isomorphism classes [V] of A-modules V modulo the relations [V] = [U] + [W] for each short exact sequence of A-modules 0 → U → V → W → 0. It is a classical fact (cf. [Ba1], p. 404) that G_{0}(A) is a free abelian group with basis given by the classes [V_{i}] (i = 1, …, t).

Similarly, K_{0}(A) denotes the Grothendieck group of the category of projective (finite dimensional left) A-modules, that is, the abelian group that is generated by the isomorphism classes [P] of projective A-modules P modulo the relations [P ⊕ Q] = [P] + [Q]. Again, K_{0}(A) is free abelian, with basis {[P_{i}]|i = 1, …, t}.

The two Grothendieck groups are related via the Cartan map

c = c_{A} : K_{0}(A) → G_{0}(A) , [P] ↦ [P] .

The matrix C ∈ M_{t}(ℤ) which represents c with respect to the above bases is called the Cartan matrix:

C = (c_{i,j})_{t×t} with c_{i,j} = multiplicity of V_{j} as composition factor of P_{i}.

1.2 Orthogonality. Putting

⟨ . , . ⟩ : K_{0}(A) × G_{0}(A) → ℤ , ⟨P, V ⟩ = dim_{𝕜} Hom_{A}(P, V) ,

one obtains a well-defined bilinear map (cf. [Ba2], §4.3; for simplicity, the parentheses [ .] are omitted). Since P_{i}/JP_{i} ≅ V_{i}, we have Hom_{A}(P_{i}, V_{j}) ≅ Hom_{A}(V_{i}, V_{j}) which yields the orthogonality relations

⟨P_{i}, V_{j}⟩ = \{ \begin{matrix}0 & if i ≠ j \\ d_{i} & if i = j \end{matrix} , where d_{i} = dim_{𝕜} End_{A}(V_{i}) .

If 𝕜 is a splitting field for A then all d_{i} = 1. Defining

{ . , . } : K_{0}(A) × K_{0}(A) → ℤ , {a, b} = ⟨a, c(b)⟩

one obtains a bilinear form on K_{0}(A) whose matrix with respect to the given basis of K_{0}(A) is

C′ = (c′_{i,j})_{t×t} where c′_{i,j} = d_{i}c_{j,i} .

Indeed, c([P_{j}]) = ∑_{ℓ} c_{j,ℓ}[V_{ℓ}] implies {P_{i}, P_{j}} = ∑_{ℓ} c_{j,ℓ}⟨P_{i}, V_{ℓ}⟩ = c′_{i,j}, by the orthogonality relations.

1.3 Trace spaces. Following [Ba2], we write

T = T_{A} : A → T (A) = A/[A, A]

for the natural projection to the quotient of A by the 𝕜-linear span [A, A] of the Lie commutators [a, b] = ab − ba in A. We let

[ .] : A → T (A)_{reg} = A/ ([A, A] + J) ≅ T (A/J)

denote the canonical map. The space of 𝕜-valued trace functions on A is the 𝕜-subspace of A^{∗} = Hom_{𝕜}(A, 𝕜) that is defined by

C(A) = {f ∈ A^{∗} : f(ab) = f(ba) for all a, b ∈ A} ≅ T (A)^{∗} .

We define the space of regular 𝕜-valued trace functions by

C(A)_{reg} = {f ∈ C(A) : f vanishes on J} ≅ C(A/J) ≅ T (A)_{reg}^{∗} .

If 𝕜 is a splitting field for A and p = char 𝕜 > 0, then [A, A]+J = {a ∈ A : a^{p^{n}} ∈ [A, A] for some n ≥ 0} (e.g., [P], p. 56), and hence

C(A)_{reg} = {f ∈ A^{∗} : f(a) = 0 for all a ∈ A with a^{p^{n}} ∈ [A, A] for some n ≥ 0} .

1.4 Hattori-Stallings ranks. Let P be a projective A-module. The trace map is defined by

tr_{P} = tr_{P/A} : End_{A}(P) ⟶^{≅} Hom_{A}(P, A) ⊗_{A} P ⟶^{τ_{P}} T (A), τ_{P} (f ⊗ v) = T (f(v))

(cf. [Ba2]). If {(f_{i}, v_{i})}_{1}^{n} ⊆ Hom_{A}(P, A) × P is a dual basis for P, that is, v = ∑_{i} f_{i}(v)v_{i} holds for all v ∈ P, then

tr_{P} (ϕ) = ∑_{i} f_{i}(ϕ(v_{i})) + [A, A] (ϕ ∈ End_{A}(P)) .

The Hattori-Stallings rank map is defined by (cf. [Ba2], §2.4)

r : K_{0}(A) → T (A), [P] ↦ r_{P} = tr_{P} (Id_{P}) .

Explicitly, writing P = A^{n} ·e for some idempotent matrix e = (e_{i,j}) ∈ M_{n}(A), we have r_{P} = ∑ T (e_{i,i}). In particular, if P is free of rank n over A then r_{P} = T (n). Since each P_{i} has the form P_{i} = Ae_{i} for some primitive idempotent e_{i} ∈ A, the image of the Hattori-Stallings rank map is exactly the additive subgroup of A/[A, A] that is generated by the (primitive) idempotents of A. Finally, we define

ρ : K_{0}(A) ⟶^{r} T (A) ↠ T (A)_{reg} ,

where the last map is the canonical epimorphism A/[A, A] ↠ A/ ([A, A] + J), and we put

ρ_{i} = ρ([P_{i}]) = [e_{i}] ∈ T (A)_{reg} (i = 1, …, t).

1.5 Characters. Let V be an A-module and denote its structure map A → End_{𝕜}(V) by a ↦ a_{V} (a ∈ A). Then the character χ_{V} of V is defined by

χ_{V} (a) = tr_{V/𝕜}(a_{V}) ∈ 𝕜 (a ∈ A) .

The characters χ_{i} = χ_{V_{i}} are called the irreducible characters of A. Each character χ_{V} is an element of C(A) and if 0 → U → V → W → 0 is a short exact sequence of A-modules then χ_{V} = χ_{U} + χ_{W}. In particular, since the irreducible characters clearly belong to C(A)_{reg}, so do all characters χ_{V}, and the character map

χ : G_{0}(A) → C(A)_{reg}, [V] ↦ χ_{V}

is a well-defined group homomorphism which satisfies χ_{V} (1) = dim_{𝕜}(V).

1.6 Theorem. (a) The following diagrams commute.

K_{0}(A) × G_{0}(A) ⟶^{⟨ . , . ⟩} ℤ ^{ρ×χ}↓  ↓^{can.} T (A)_{reg} × C(A)_{reg} ⟶_{evaluation} 𝕜 and  \begin{matrix}K_{0}(A) & ⟶^{c_{A}} & G_{0}(A) \\ ^{ρ}↓ & & ↓^{χ} \\ T (A)_{reg} & ⟶_{( .)^{t}} & C(A)_{reg} \end{matrix} .

Here, the map ρ is as in §1.4 and ( .)^{t} : T (A)_{reg} → C(A)_{reg} = T (A)_{reg}^{∗} is defined by

([a])^{t}([b]) = tr_{A/𝕜}(L_{b} ∘ R_{a}) (a, b ∈ A),

where L_{b}, R_{a} ∈ End_{𝕜}(A) are left multiplication with b and right multiplication with a, respectively.

(b) The nonzero irreducible characters χ_{i} ∈ C(A)_{reg} are linearly independent over 𝕜. If 𝕜 is a splitting field for A then the χ_{i} (i = 1, …, t) form a 𝕜-basis of C(A)_{reg} and χ induces an isomorphism of 𝕜-vector spaces

χ̃ = Id_{𝕜} ⊗χ : { \atop G_{0}(A)} = 𝕜 ⊗_{ℤ} G_{0}(A) ⟶^{≅} C(A)_{reg} .

(c) If 𝕜 is a splitting field for A then the ρ_{i} (i = 1, …, t) form a 𝕜-basis of T (A)_{reg} and ρ induces an isomorphism of 𝕜-vector spaces

ρ̃ = Id_{𝕜} ⊗ρ : { \atop K_{0}(A)} = 𝕜 ⊗_{ℤ} K_{0}(A) ⟶^{≅} T (A)_{reg} .

Proof. The diagrams in (a) are minor modifications of [Ba2], 4.3 and 4.7, respectively. To see that ( .)^{t} is a well-defined map, note that tr_{A/𝕜}(L_{b} ∘ R_{a}) = 0 if a ∈ J or a ∈ [A, A], because L_{b} ∘ R_{a} = R_{a} ∘ L_{b} is a nilpotent endomorphism in the former case and a commutator of endomorphisms in the latter. Similarly for b in place of a.

(b) Assume first that 𝕜 is a splitting field for A. Then all d_{i} = 1 in the orthogonality relations in §1.2, and the first diagram in (a) yields χ_{i}(ρ_{j}) = δ_{i,j}1_{𝕜}. Thus, if ∑ k_{i}χ_{i} = 0 for k_{i} ∈ 𝕜 then 0 = ∑ k_{i}χ_{i}(ρ_{j}) = k_{j} for all j, thereby proving linear independence of the χ_{i}’s. Furthermore, the number, t, of non-isomorphic irreducible A-modules, equals the number of matrix components of A/J which in turn is equal to the dimension of T (A/J) ≅ C(A)_{reg}^{∗}. Hence t = dim_{𝕜} C(A)_{reg}, and so the χ_{i} span C(A)_{reg}.

For independence of the nonzero irreducible characters χ_{i} in the general case, choose a field extension 𝕂/𝕜 so that 𝕂 is a splitting field for Ã = 𝕂 ⊗ A and let χ̃_{i} ∈ C(Ã) = 𝕂 ⊗ C(A) denote the character of the Ã-module Ṽ_{i} = 𝕂 ⊗ V_{i}. So χ̃_{i} = 1 ⊗ χ_{i}. For i ≠ j, the Ã-modules Ṽ_{i} and Ṽ_{j} have no common composition factor ([CR], p. 170, Exercise 7.9). Therefore, writing each χ̃_{i} in terms of irreducible characters of Ã, there is no overlap between the various i’s. In view of the foregoing, this implies that the nonzero χ̃_{i}’s are independent over 𝕂, and hence the nonzero χ_{i}’s are independent over 𝕜.

(c) The proof of (c) is dual to the first part of the proof of (b). □

1.7 Symmetric algebras. The 𝕜-algebra A is called symmetric if there exits a nondegenerate 𝕜bilinear form

β : A × A → 𝕜

which is associative and symmetric, that is, β(ab, c) = β(a, bc) and β(a, b) = β(b, a) holds for all a, b, c ∈ A. We recall some well-known facts about symmetric algebras:

(1) Group algebras of finite groups and all finite dimensional semisimple algebras are symmetric.

(2) If A is symmetric then, for any irreducible A-module V, the socle of P (V) is isomorphic to V. In particular, it follows that P (V^{∗}) ≅ P (V)^{∗} as (right) A-modules. (In case A is a Hopf algebra, the switch from left to right modules is unnecessary, because the side of the action is preserved for duals by means of the antipode. See §2.1 below.)

(3) If A is symmetric then the matrix C′ = (c′_{i,j}) of §1.2 is symmetric and, consequently, the form

{ . , . } is symmetric. Facts (1) and (2) can be found in [CR], p. 198ff, and (3) is [La], Theorem 9.8 (at least for 𝕜 large enough; the general case follows along the same lines). Part (b) of the following Lemma, apparently a well known fact, has been pointed out to us by S. Montgomery and H.-J. Schneider who also provided the proof given below. It is included here for lack of a suitable reference.

Lemma. Let A is a symmetric algebra with form β. Then:

(a) β([a, b], c) = β(a, [b, c]) holds for all a, b, c ∈ A. In particular, C(A) ≅ Z(A), the center of A, and C(A)_{reg} ≅ ann_{Z(A)}(J) as 𝕜-vector spaces.

(b) If γ : A × A → 𝕜 is any nondegenerate bilinear form which is associative then there exists a unit u ∈ A such that γ(a, b) = γ(u^{−1}bu, a) holds for all a, b ∈ A.

Proof. (a) First, β(ba, c) = β(c, ba) = β(cb, a) = β(a, cb) and so β([a, b], c) = β(ab, c) − β(ba, c) = β(a, bc) − β(a, cb) = β(a, [b, c]). Thus, a ∈ Z(A) if and only if β(a, .) vanishes on [A, A], whence the first isomorphism follows. For the second isomorphism, observe that β(a, J) = 0 is equivalent with β(aJ, A) = 0, and hence with aJ = 0.

(b) Viewing A^{∗} as left A-module via (af)(b) = f(ba) for f ∈ A^{∗} and a, b ∈ A, the form γ determines a left A-isomorphism G : A → A^{∗}, G(a) = γ( . , a). Letting B : A → A^{∗} denote the analogous isomorphism corresponding to β, we obtain a left A-isomorphism ϕ = B^{−1} ∘ G : A → A. Thus ϕ(a) = au, where u = ϕ(1) is a unit of A, and so G(1) = B(u). Now, for a, b ∈ A,

γ(a, b) = γ(ab, 1) = G(1)(ab) = B(u)(ab) = β(ab, u)

and, consequently, γ(u^{−1}bu, a) = β(u^{−1}bua, u). Finally, using associativity and symmetry of β, one computes β(u^{−1}bua, u) = β(u, u^{−1}bua) = β(bu, a) = β(a, bu) = β(ab, u) which entails the claimed identity γ(a, b) = γ(u^{−1}bu, a). □

2. Dimensions and Symmetry

Throughout this section, H denotes a finite dimensional Hopf algebra over 𝕜, with counit ε, antipode S, and comultiplication Δ. The latter will be written Δ(h) = ∑ h_{1} ⊗ h_{2} for h ∈ H. Recall that all H-modules are left modules and are assumed to be finite dimensional over 𝕜.

2.1 Homomorphisms. Let V and W be H-modules. Then Hom_{𝕜}(V, W) can be made into an Hmodule by defining

(hf)(v) = ∑ h_{1}f(S(h_{2})v) (h ∈ H, v ∈ V, f ∈ Hom_{𝕜}(V, W)).

In the special case where W = 𝕜 = 𝕜_{ε} is the trivial H-module, so H acts on 𝕜 via the counit ε, this simplifies to the following formula describing the H-action on V^{∗} = Hom_{𝕜}(V, 𝕜):

(hf)(v) = f(S(h)v) (h ∈ H, v ∈ V, f ∈ V^{∗}).

Viewing tensor products of H-modules as H-modules by means of the diagonal map Δ, the canonical isomorphisms (cf. [Bou], p. II.77 and II.80)

W ⊗ V^{∗} ⟶^{≅} Hom_{𝕜}(V, W) ,  w ⊗ f ↦ (v ↦ f(v)w)
V^{∗} ⊗ W^{∗} ⟶^{≅} (W ⊗ V)^{∗} ,  f ⊗ g ↦ (w ⊗ v ↦ g(w)f(v)) .

are in fact an H-module isomorphisms. Finally, the space of H-invariants Hom_{𝕜}(V, W)^{H} = {f ∈ Hom_{𝕜}(V, W) : hf = ε(h)f for all h ∈ H} coincides with the space of H-module maps Hom_{H}(V, W) ([L], Proposition 2.3 or [Zhu], Lemma 1).

2.2 Duality. Let V be a H-module. Viewing V^{∗∗} as H-module as in §2.1, the canonical isomorphism ψ : V → V^{∗∗}, ψ(v)(f) = f(v) satisfies ψ(hv) = S^{−2}(h)ψ(v). Therefore, as H-modules,

V^{∗∗} ≅ V^{(S^{−2})} ,

the S^{−2}-twist of V. Since twists by inner automorphisms do not affect the isomorphism type, we see in particular:

If S^{2} is inner then V^{∗∗} ≅ V holds for all H-modules V.

Using ( .)^{∗} to denote transpose maps, we have an isomorphism of 𝕜-vector spaces

( .)^{∗} : Hom_{H}(V, W) ⟶^{≅} Hom_{H}(W^{∗}, V^{∗}) .

Thus ( .)^{∗} becomes an exact contravariant automorphism of the category of finite dimensional Hmodules. Clearly, ( .)^{∗} respects direct sums, and some power of ( .)^{∗} is equivalent to the identity, because S has finite order ([R3]). In particular, V^{∗} is irreducible (resp., indecomposable) if and only if V is.

The H-module V is called self-dual if V^{∗} ≅ V. The trivial H-module 𝕜 = 𝕜_{ε} and the regular left H-module H are self-dual, the former as a consequence of the identity ε ∘ S = ε, the latter by [Sw], Theorem 5.1.3. Consequently, projectivity also transfers between V and V^{∗}.

2.3 Theorem. Assume that H is involutory (that is, S^{2} = Id).

(a) ([L], Theorem 2.8) If H is semisimple then p does not divide the dimensions of any absolutely irreducible H-module.

(b) If H is not semisimple, then p divides the dimension of every projective H-module.

Recall that an H-module V is absolutely irreducible if for every field extension 𝕂/𝕜, 𝕂 ⊗ V is an irreducible 𝕂 ⊗ H-module or, equivalently, if End_{𝕜}(V) = 𝕜 (cf. [CR], Theorem 3.43).

Proof. Both parts follws from the observation that, for any H-module V, the trace map tr_{V/𝕜} : End_{𝕜}(V) → 𝕜 = 𝕜_{ε} of §1.4 is an H-module map. Indeed, identifying End_{𝕜}(V) with V ⊗ V^{∗} as in §2.1, it suffices to check H-linearity of the map τ : V ⊗ V^{∗} → 𝕜, v ⊗ f ↦ f(v). Using the identity ∑ S(h_{2})h_{1} = ε(h)1_{H} for h ∈ H (from S^{2} = Id) we compute

τ(h · v ⊗ f) = τ(∑ h_{1}v ⊗ h_{2}f) = ∑(h_{2}f)(h_{1}v) = ∑ f(S(h_{2})h_{1}v) = ε(h)f(v) = h · τ(v ⊗ f) ,

as required.

(a) Since H is semisimple, there is a left integral t in H with ε(t) ≠ 0 in 𝕜. Note that t · End_{𝕜}(V) consists of H-invariants and hence is contained in End_{H}(V), by §2.1. Thus, if f ∈ End_{𝕜}(V) and V is absolutely irreducible, then tf = c_{f} Id_{V} for some c_{f} ∈ 𝕜, and hence

c_{f} dim_{𝕜} V = tr_{V/𝕜}(tf) = t tr_{V/𝕜}(f) = ε(t) tr_{V/𝕜}(f) .

Choosing f ∈ End_{𝕜}(V) with tr_{V/𝕜}(f) = 1, we deduce that dim_{𝕜} V ≠ 0 in 𝕜.

(b) Now let P be a projective H-module and assume, by way of contradiction, that p does not divide dim_{𝕜} P. Then the trace map tr_{P/𝕜} splits via σ : 𝕜_{ε} → End_{𝕜}(P), k ↦ (dim_{𝕜} P)^{−1}k · Id_{P}. Therefore, 𝕜_{ε} is isomorphic to a direct summand of End_{𝕜}(P). Inasmuch as End_{𝕜}(P) ≅ P ⊗ P^{∗} is projective, by the Fundamental Theorem of Hopf Modules ([Mo], Theorem 1.9.4.), we conclude that 𝕜_{ε} is projective as well. This forces H to be semisimple, contrary to our assumption. □

Remarks. The semisimplicity hypothesis in (a) is definitely necessary. Indeed, the restricted enveloping algebra H = u(𝔤) of the restricted p-Lie algebra 𝔤 = sl(2, 𝕜) over a field 𝕜 of characteristic p > 2 has an irreducible module of dimension p. See §4.4 below for a detailed discussion of this example. Part (b) implies in particular the known result ([L], Theorem 4.3) that any involutory Hopf algebra H with dim_{𝕜} H not divisible by p is semisimple. Since this is false in general if H is not involutory (cf. §4.1), this hypothesis is necessary for (b) to hold. In (a), semisimplicity of H may very well entail that H is involutory (Kaplansky’s conjecture [K]; known to be true in characteristic 0 [LR2]. For an alternative proof, see [PQ2]).

2.4 Local Hopf subalgebras. Stronger estimates than the one provided by Theorem 2.3(b) are sometimes possible by using local Hopf subalgebras. These are exactly those Hopf subalgebras L of H such that the augmentation ideal L^{+} = Ker ε_{L} is nilpotent.

Lemma. Let L be a local Hopf subalgebra of H. Then dim_{𝕜} L divides the dimension of every projective H-module.

Proof. Since H is free as (left and right) L-module, by the Nichols-Zoeller Theorem ([Mo], Theorem 3.1.5), every projective H-module is projective, and hence free, over L. □

Remarks and Examples. If H = 𝕜G is the group algebra of a finite group G then the local Hopf subalgebras of H are the group subalgebras 𝕜P, where P is a p-subgroup of G (0-subgroup means ⟨1⟩). The local Hopf subalgebras of the restricted enveloping algebra H = u(𝔤) of a finite dimensional restricted p-Lie algebra (𝔤, [p]) over a field 𝕜 of characteristic p > 0 are the enveloping subalgebras u(𝔭), where 𝔭 is a p-nilpotent p-Lie subalgebra of 𝔤. Here, 𝔭 is p-nilpotent if some power of the p-map [p] is 0 on 𝔭. If H is a semisimple Hopf algebra then all Hopf subalgebras are semisimple as well ([Mo], Corollary 3.2.3). Therefore, 𝕜 is the only local Hopf subalgebra in this case. Even if H is not semisimple, local Hopf subalgebras ≠ 𝕜 need not exist; the Hopf algebra H_{4} in characteristic p ≠ 2 is an example. (Note, however, that H_{4} is free over the local non-Hopf subalgebra L = 𝕜[x], in the notation of §4.1, and so the conclusion of the Lemma applies to L as well.)

2.5 Projective covers, unimodularity, and symmetry. Recall that if V is an irreducible Hmodule then the projective cover P (V) is the unique (up to isomorphism) indecomposable projective H-module which maps onto V (cf. [CR], p. 131). In the following Lemma, we let α ∈ H^{∗} denote the “distinguished” group-like element of H^{∗} such that

th = α(h)t

holds for every left integral t in H and every h ∈ H (cf. [Mo], p. 22). The Hopf algebra H is unimodular if every left integral is a right integral for H or, equivalently, if α = ε.

Lemma. P (𝕜_{ε})^{∗} ≅ P (𝕜_{α}). In particular, P (𝕜_{ε}) is self-dual if and only if H is unimodular.

Proof. Put Q = P (𝕜_{ε})^{∗}, an indecomposable projective H-module, by §2.2. It suffices to show that Q maps onto 𝕜_{α}. But Q ≅ He for some primitive idempotent e ∈ H, and He contains a nonzero left integral t of H, because 𝕜_{ε} ≅ 𝕜_{ε}^{∗} embeds into Q. Thus t = te = α(e)t, and so α(e) ≠ 0 and

Q ⟶^{≅} He ⟶^{α} 𝕜_{α} is epi. □

Proposition ([OS]). H is a symmetric algebra precisely if H is unimodular and S^{2} is inner.

Proof. The conditions are sufficient. For, if λ ∈ H^{∗} is a nonzero right integral then, putting (h, k) = λ(hk) for h, k ∈ H, one obtains an associative bilinear form ( . , .) : H × H → 𝕜 which is well-known to be nondegenerate ([LS], Theorem or [R], Cor. 2(b)). Moreover, since H is unimodular, the form also satisfies (h, k) = (S^{2}(k), h) for h, k ∈ H ([LS], Proposition 8 or [R], Theorem 3(a)). Thus, choosing a unit u ∈ H so that S^{2}(h) = uhu^{−1} holds for all h ∈ H, we can define β : H×H → 𝕜 by β(h, k) = (uh, k) to obtain the required symmetric form.

Conversely, assume that H is symmetric. Then P (V^{∗}) ≅ P (V)^{∗} holds for all irreducible H-modules V, by §1.7 (2). In particular, the above Lemma implies that H is unimodular. (For a more elementary proof of this implication, see [H2], Theorem 2.) Thus the associative nondegenerate form (h, k) = λ(hk) used in the first paragraph of the proof satisfies (h, k) = (S^{2}(k), h). On the other hand, by Lemma 1.7(b), (h, k) = (u^{−1}ku, h) for some unit u ∈ H. Consequently, S^{2}(k) = u^{−1}ku which shows that S^{2} is inner. □

Remarks and Examples. (1) Finite dimensional commutative Hopf algebras are symmetric, since they are unimodular and involutory. Cocommutative Hopf algebras are symmetric precisely if they are unimodular which is not generally the case (cf. (4) below). Thus symmetry does not pass from a Hopf algebra to its dual. (See also (3) below.)

(2) The fact that S^{2} is inner for symmetric Hopf algebras H entails in particular that V^{∗∗} ≅ V holds for all H-modules V (§2.2). Furthermore, in view of §1.7 (1), the Proposition implies that

S^{2} is inner if H is semisimple.

For 𝕜 algebraically closed, this fact has been noted by Larson ([L], Prop. 3.5, see also [R], Theorem 5); the general case is due to Oberst and Schneider [OS]. We also remark that Drinfeld has shown that S^{2} is inner for any almost cocommutative Hopf algebra ([Dr], cf. also [Mo], Proposition 10.1.4).

(3) The Drinfeld double D(H) of any Hopf algebra H is always symmetric, because the conditions of the Proposition are satisfied by [R2], Theorem 4 and Cor. 2. (For the fact that S^{2} is inner, one could also quote Drinfeld’s more general theorem about almost commutative Hopf algebras mentioned in (2) above.) Therefore, any H embeds in a symmetric Hopf algebra, and symmetry is not in general inherited by Hopf subalgebras. Moreover, the dual D(H)^{∗} is unimodular precisely if both H and H^{∗} are, and similarly for the property of the square of the antipode being inner ([R2], Cor. 4 and Prop. 8). Consequently, D(H)^{∗} is symmetric if and only if both H and H^{∗} are.

(4) Let H = u(𝔤) denote the restricted enveloping algebra of the finite dimensional restricted p-Lie algebra (𝔤, [p]) over a field 𝕜 of characteristic p > 0, and let ad_{𝔤}(x) ∈ End_{𝕜}(𝔤) be defined as usual by ad_{𝔤}(x)(y) = [x, y] for x, y ∈ 𝔤. By [LS], Cor. on p. 91 (cf. also [Sch]), symmetry (or unimodularity) of H is equivalent to

tr(ad_{𝔤}(x)) = 0 for all x ∈ 𝔤.

This condition is certainly satisfied if 𝔤 is nilpotent or if 𝔤 = [𝔤, 𝔤] and so, in particular, if 𝔤 is simple. The condition is also satisfied for the diamond algebra, for example, but not for the 2-dimensional solvable p-Lie algebra (cf. §4.3).

3. The Grothendieck ring

The notations introduced at the beginning of §2 remain in effect. Furthermore, as in §1, V_{1}, …, V_{t} denotes a full set of nonisomorphic irreducible H-modules, P_{i} = P (V_{i}) are their projective covers, and J = rad H is the Jacobson radical of H.

3.1 The Grothendieck ring. As is well-known (e.g., [Ser]), the Grothendieck group G_{0}(H) is endowed with the structure of an augmented ℤ-algebra with identity element 1 = [𝕜_{ε}]: The augmentation is dim_{k} : G_{0}(H) → ℤ and multiplication is given by [V] · [W] = [V ⊗ W]. We will refer to G_{0}(H) as the Grothendieck ring of H. Putting [V]^{∗} = [V^{∗}] we obtain an anti-automorphism ∗ of the ring G_{0}(H) which permutes the basis {[V_{i}]} of G_{0}(H), and ∗ is an involution of G_{0}(H) if S^{2} is inner (§§2.1, 2.2).

Similarly, [P]^{∗} = [P^{∗}] yields an automorphism of the group K_{0}(H) which permutes the basis {[P_{i}]} and has order 2 if S^{2} is inner. Putting [P] · [V] = [P ⊗ V] induces a right G_{0}(H)-module structure on K_{0}(H). The fact that P ⊗ V is projective if P is can be seen as follows. Clearly, it suffices to prove that H ⊗ V is projective for any V. Now H ⊗ V is a (left) H-Hopf module (cf. [Mo], Example 1.9.3), and so H ⊗ V is in fact free as left H-module, by the Fundamental Theorem of Hopf Modules ([Mo], Theorem 1.9.4). — The Cartan map c : K_{0}(H) → G_{0}(H) is a ∗-equivariant map of G_{0}(H)-modules.

With respect to the G_{0}(H)-modules structures and duality maps ∗ on K_{0}(H) and G_{0}(H), the bilinear maps ⟨ . , . ⟩ : K_{0}(A) × G_{0}(A) → ℤ and { . , . } : K_{0}(H) × K_{0}(H) → ℤ of §1.2 satisfy the identities

⟨ax, y⟩ = ⟨a, yx^{∗}⟩ , {a, b} = {b^{∗}, a^{∗}} for x, y ∈ G_{0}(H) and a, b ∈ K_{0}(H).

The first follows from the 𝕜-linear isomorphisms Hom_{H}(P ⊗V, W) ≅ (W ⊗ V^{∗} ⊗ P^{∗})^{H} ≅ Hom_{H}(P, W ⊗ V^{∗}), and the second from the isomorphism Hom_{H}(V, W) ≅ Hom_{H}(W^{∗}, V^{∗}) (cf. §§2.1, 2.2). We point out two consequences of these formulas.

• The G_{0}(H)-module K_{0}(H) is faithful: Indeed, if x ∈ G_{0}(H) satisfies K_{0}(H)x = 0 then 0 = ⟨K_{0}(H)x, 1⟩ = ⟨K_{0}(H), x^{∗}⟩. In view of the orthogonality relations, the latter condition is equivalent with x^{∗} = 0, and hence with x = 0.

• The left and right radical of the form { . , . } are both equal to the kernel of the Cartan map c: Since Ker(c) is ∗-invariant and {a, b} = {b^{∗}, a^{∗}} holds for all a, b ∈ K_{0}(H), it suffices to show that the right radical of { . , . } equals Ker(c). But, as above, {K_{0}(H), a} = ⟨K_{0}(H), c(a)⟩ = 0 is equivalent to c(a) = 0.

Recall also that the form { . , . } has matrix C′, and it is symmetric if H is symmetric.

Remarks and Examples. (1) If I is an ideal of H that is contained in J = rad H, then inflation of modules along the canonical map π : H → H/I gives a group isomorphism G_{0}(H/I) →^{≅} G_{0}(H) (cf. [Ba1], p. 455). In case I is a Hopf ideal of H, this is an isomorphism of rings. This applies in particular if J is a Hopf ideal of H. See §3.3 below for a detailed discussion of this case.

(2) If H is semisimple then all H-modules are projective. Thus K_{0}(H) = G_{0}(H) and the Cartan map c is the identity map in this case. For example, let H = (𝕜G)^{∗} denote the dual of the group algebra 𝕜G of the finite group G, so H ≅ 𝕜^{|G|} as 𝕜-algebras. Here, G_{0}(H) ≅ ℤG, the integral group ring of G. Up to a finite separable field extension, this example covers all semisimple commutative Hopf algebras (cf. [Mo], Theorem 2.3.1).

(3) Let H be almost cocommutative in the sense of Drinfeld [Dr]. Then S^{2} is inner and V ⊗ W ≅ W ⊗ V holds for all H-modules V and W ([Dr], cf. also [Mo], Lemma 10.1.2 and Proposition 10.1.4). Hence G_{0}(H) is a commutative ring and ∗ is an involution in this case.

3.2 Semiprimeness. Recall a subgroup X of K_{0}(H) is said to be isotropic with respect to form { . , . } if {X, X} = 0. In case { . , . } is symmetric, this condition is equivalent with {x, x} = 0 for all x ∈ X.

Proposition. K_{0}(H) contains no nonzero ∗-invariant isotropic G_{0}(H)-submodules if and only if G_{0}(H) is a semiprime ring and the Cartan map c is injective.

Proof. First assume that K_{0}(H) contains no nonzero ∗-invariant isotropic G_{0}(H)-submodule. Then, in particular, Ker(c) = 0 and so c is injective. Let N denote the nilpotent radical of the (Noetherian) ring G_{0}(H) and suppose, by way of contradiction, that N ≠ 0. Choose n so that I = N^{n} ≠ 0, but I^{2} = 0, and note that I is ∗-invariant, because N certainly is. Also, since c is injective, c ⊗_{ℤ} ℚ is bijective, and so I ∩ Im(c) ≠ 0. Consequently, letting X denote the preimage of I in K_{0}(H) under c, we have I ⊇ c(X) and X is a nonzero ∗-invariant G_{0}(H)-submodule of K_{0}(H). Since I^{2} ⊇ c(XI), we deduce that c(XI) = 0, and hence XI = 0. On the other hand, ⟨XI, 1⟩ = ⟨X, I⟩ ⊇ {X, X} ≠ 0, by assumption on the form { . , . }. This contradiction shows that we must have N = 0, and so G_{0}(H) is semiprime.

Conversely, assume that G_{0}(H) is a semiprime ring and the Cartan map c is injective. If X is a nonzero ∗-invariant isotropic G_{0}(H)-submodule of K_{0}(H), then I = c(X) is a nonzero ∗-invariant right ideal of G_{0}(H) (and so I is actually a two-sided ideal of G_{0}(H)). Moreover, 0 = {X, X} = ⟨X, I⟩ = ⟨XI, 1⟩ = ⟨XIG_{0}(H), 1⟩ = ⟨XI, G_{0}(H)⟩, and hence the orthogonality relations imply that XI = 0. Therefore, I^{2} = c(XI) = 0, contradicting semiprimeness of G_{0}(H). □

Examples. (1) Semisimple Hopf algebras H: In this case, C is the identity matrix and C′ is a diagonal matrix with positive entries. Therefore the quadratic form Q(x) = {x, x} is positive definite and no nonzero isotropic submodules of K_{0}(H) can exist. Therefore, G_{0}(H) is semiprime if H is semisimple. — More generally, this shows that G_{0}(H) is a semiprime ring if J is a Hopf ideal of H, because G_{0}(H) ≅ G_{0}(H/J) holds in this case.

(2) Group algebras over sufficiently large fields: For group algebras 𝕜G of finite groups G, the Grothendieck ring G_{0}(𝕜G) can be explicitly described in terms of (Brauer or ordinary) characters, and semiprimeness is evident from this description. However, it also follows from the Proposition, in view of the fact that the Cartan matrix C (= C′, because 𝕜 is assumed large) is known to be invertible and to have the form C = D^{T} D, where D is an integer matrix, the so-called decomposition matrix (cf. [CR], Cor. 18.10 and Theorem 18.25). Consequently, C is positive definite, and semiprimeness of G_{0}(𝕜G) follows as in (1).

(3) The restricted enveloping algebra H = u(sl(2, 𝕜)) provides an example of a Grothendieck ring G_{0}(H) which is not semiprime. See §4.4.

3.3 The case when J = rad H is a Hopf ideal. We discuss the case where J = rad H is a Hopf ideal of H in some detail. Some properties of this case have already been noted in §3.1 Remarks and Examples (1) and in §3.2 Examples (1).

Since ε(J) = 0 and S(J) ⊆ J are automatic, J = rad H is a Hopf ideal if and only if

Δ(J) ⊆ H ⊗ J + J ⊗ H .

Several further equivalent conditions are given in the following Lemma. In this connection, see also [PQ] and the remarks in [Mo], p. 62/63.

Lemma. The following conditions are equivalent.

(1) J is a Hopf ideal of H.

(2) Tensor products of semisimple H-modules are semisimple.

(3) If V, W are H-modules, then J · (W ⊗ V) ⊆ JW ⊗ V + W ⊗ JV.

(4) If V, W are H-modules, with V semisimple, then J · (W ⊗ V) = JW ⊗ V.

(5) Same as (4), but with V semisimple and W projective.

(6) If V, W are H-modules, with V semisimple, then P (W ⊗ V) ≅ P (W) ⊗ V.

(7) Same as (6), but with V and W both semisimple.

Proof. (1) ⇔ (2): (2) is equivalent with the equation J · (H/J ⊗ H/J) = 0 which in turn is equivalent with Δ(J) ⊆ H ⊗ J + J ⊗ H.

(1) ⇒ (3) is clear, and (3) ⇒ (1) follows by taking V = W = H.

(3) ⇒ (5): Write W ⊕ Q ≅ H^{r} for suitable Q and r. Then we have H-module isomorphisms

(W ⊗ V) ⊕ (Q ⊗ V) ≅ H^{r} ⊗ V ≅ H^{r dim_{𝕜} V} ,

the last isomorphism being a consequence of the Fundamental Theorem of Hopf Modules (cf. §3.1). Hence, as 𝕜-vector spaces,

J · (W ⊗ V) ⊕ J · (Q ⊗ V) ≅ J^{r dim_{𝕜} V} ≅ J^{r} ⊗ V ≅ (JW ⊕ JQ) ⊗ V ≅ (JW ⊗ V) ⊕ (JQ ⊗ V) .

But (3) implies that J · (W ⊗ V) ⊆ JW ⊗ V, and similarly for Q, because V is semisimple. Thus, comparing dimensions, we obtain (5).

(5) ⇒ (4) and (6): We write head X = X/JX for all H-modules X. Now let V, W be H-modules, with V semisimple. Then head P (W) ≅ head W, a general property of projective covers, and (5) implies that head(P (W) ⊗ V) ≅ (head P (W)) ⊗ V. Thus (head W) ⊗ V ≅ head(P (W) ⊗ V) is semisimple, and so head(W ⊗ V) canonically maps onto (head W) ⊗ V. On the other hand, the canonical epimorphism P (W) ⊗ V ↠ W ⊗ V entails an epimorphism (head W) ⊗ V ≅ head(P (W) ⊗ V) ↠ head(W ⊗ V). So (head W) ⊗ V and head(W ⊗ V) are in fact isomorphic which proves (4). For (6), consider the composite epimorphism

P (W ⊗ V) ↠ W ⊗ V ↠ head(W ⊗ V) ≅ (head W) ⊗ V ≅ head(P (W) ⊗ V) ,

where the first two epimorphisms are the canonical ones and the last isomorphism was established above. Using the Nakayama Lemma we deduce that P (W ⊗ V) maps onto P (W) ⊗ V. Similarly, the canonical epimorphism P (W) ⊗ V ↠ W ⊗ V implies that P (W) ⊗ V maps onto P (W ⊗ V). Inasmuch as P (W) ⊗ V and P (W ⊗ V) are both finite dimensional, the latter two epimorphism are in fact isomorphisms, thereby proving (6).

(6) ⇒ (7) is trivial.

(7) ⇒ (2): Let V and W be semisimple H-modules and put X = W ⊗ V. By two applications of (7),

P (X) ≅ P (W) ⊗ V ≅ P (𝕜_{ε}) ⊗ X .

On the other hand, by general properties of projective covers, P (X) ≅ P (head X), and (7) applied to head Xfurther implies that P (X) ≅ P (𝕜_{ε}) ⊗ head X. Therefore, P (𝕜_{ε}) ⊗ X ≅ P (𝕜_{ε}) ⊗ head X. Comparing dimensions, we conclude that the canonical map X ↠ head X is an isomorphism and so X is semisimple.

Since (4) clearly implies (5), the proof is complete. □

Corollary. Assume that J = rad H is a Hopf ideal of H. Then, for any H-module V, the projective cover P (V) is given by P (V) ≅ P (𝕜_{ε}) ⊗ (V/JV). Consequently, K_{0}(H) is a free G_{0}(H)-module of rank 1 with generator [P (𝕜_{ε})]. Furthermore, dim_{𝕜} P (𝕜_{ε}) is an eigenvalue of the Cartan matrix C, and hence a divisor of det C in ℤ.

Proof. In view of the general isomorphism P (V) ≅ P (V/JV), the isomorphism P (V) ≅ P (𝕜_{ε}) ⊗ (V/JV) follows from (7) in the Lemma applied to V/JV.

In particular, we have P_{i} ≅ P (𝕜_{ε}) ⊗ V_{i} for i = 1, …, t which can be written as a matrix equation over G_{0}(H) as follows:

[P (𝕜_{ε})] · ( [V_{1}] ; ⋮ ; [V_{t}] ) = C · ( [V_{1}] ; ⋮ ; [V_{t}] ) .

The assertion concerning dim_{𝕜} P (𝕜_{ε}) now follows by applying the augmentation dim_{𝕜} : G_{0}(H) → ℤ to this equation. □

Remarks and Examples. (1) If all simple H-modules are 1-dimensional (equivalently, H/J ≅ 𝕜^{r} as 𝕜-algebras for some r) then all tensor products of simple H-modules are 1-dimensional as well, and hence condition (2) of the Lemma is clearly satisfied. Thus J is a Hopf ideal in this case. By §3.1 Remarks and Examples (1) and (2), we conclude that

G_{0}(H) ≅ ℤG ,

where G is the group of group-like elements of H^{∗}. For explicit examples of this kind, see §§4.1, 4.2, 4.3 below.

(2) If H = 𝕜G is a finite group algebra then J is a Hopf ideal precisely if G has a normal Sylow p-subgroup ([M]). Furthermore, Brockhaus ([Br]) has shown that if P (V) ≅ P (𝕜_{ε}) ⊗ V holds for all semisimple 𝕜G-modules V then J is a Hopf ideal of 𝕜G. This is a substantial strengthening of the implication (6) ⇒ (1) in the Lemma for group algebras. I do not know to what extent this fact generalizes to Hopf algebras. The present proof of Brockhaus’ theorem depends on the classification of finite simple groups.

(3) The example H = u(sl(2, 𝕜)) shows that all assertions of the Corollary are false in general: K_{0}(H) is no longer generated by [P (𝕜_{ε})] in this case, and dim_{𝕜} P (𝕜_{ε}) = 2p is not an eigenvalue of the Cartan matric C. See §4.4.

3.4 The Cartan matrix. In this subsection, we determine the rank of the 𝕜-linear map

c̃_{H} = Id_{𝕜} ⊗c_{H} : { \atop K_{0}(H)} = 𝕜 ⊗_{ℤ} K_{0}(H) → { \atop G_{0}(H)} = 𝕜 ⊗_{ℤ} G_{0}(H) .

Note that rank c̃_{H} equals the rank of the Cartan matrix C of H in case p = 0, and the rank of the reduction of C mod p when p > 0. We will need the right adjoint action ◁ of H on H which is defined by

h ◁ k = ∑ S(k_{2})hk_{1} (h, k ∈ H) .

Theorem. Suppose that 𝕜 is a splitting field for H. Then

rank c̃_{H} = dim_{𝕜}(H ◁ t) ,

where t is any nonzero left integral of H. Moreover, if S^{2} is inner then the following are equivalent. (1) H is semisimple;

(2) C = Id;

(3) p does not divide det C.

Note that, for p = 0, condition (3) just says that det C ≠ 0.

Proof. By Theorem 1.6, we have a commutative diagram

\begin{matrix}{ \atop K_{0}(H)} & \underrightarrow{c̃_{H}} & { \atop G_{0}(H)} \\ ρ̃↓≅ & & χ̃↓≅ \\ T (H)_{reg} & ⟶_{( .)^{t}} & C(H)_{reg} \end{matrix} .

So rank c̃_{H} = rank( .)^{t}. By [R], Proposition 7, the map ( .)^{t} has the following description: Let t be a left integral for H and λ a right integral for H^{∗} such that λ(t) = 1. Then, for any h ∈ H,

([h])^{t} = (h ◁ t) ⇀ λ ,

where ⇀ is defined by (h ⇀ f)(k) = f(kh) for h, k ∈ H and f ∈ H^{∗}. Furthermore, λ is a free H-basis of (H^{∗}, ⇀) ([LS], p. 83 or [R], Corollary 2). Therefore, rank( .)^{t} = dim_{𝕜}(H ◁ t), as claimed.

The implications (1) ⇒ (2) ⇒ (3) are trivial. For (3) ⇒ (1), note that (3) says that c̃_{H} is injective, and hence so is the map ( .)^{t} in the above diagram. Now suppose that S^{2} is inner, say S^{2}(h) = uhu^{−1} for some unit u ∈ H. Since ε(u) ≠ 0, we have [u] ≠ 0 in T (H)_{reg}, and so ([u])^{t} ≠ 0. Therefore, by the above formula for ( .)^{t}, u ◁ t ≠ 0. The computation

u ◁ t = ∑ S(t_{2})ut_{1} = ∑ S(t_{2})S^{2}(t_{1})u = S(∑ S(t_{1})t_{2})u = ε(t)u

now shows that ε(t) ≠ 0, and so H is semisimple. □

Remarks and Examples. (1) Finite group algebras H = 𝕜G: Taking t = ∑_{g∈G} g we have g ◁ t = |ℂ_{G}(g)|𝔠_{g} for g ∈ G, where ℂ_{G}(g) denotes the centralizer of g in G and 𝔠_{g} ∈ 𝕜G is the class sum of g, that is, the sum of the elements in the conjugacy class of g. Therefore, 𝕜G ◁ t is the 𝕜-linear span of all class sums 𝔠_{g} such that p does not divide |ℂ_{G}(g)|. (These classes are in particular p-regular, that is, they consist of elements having order not divisible by p.) Consequently,

rank c̃_{𝕜G} = #{conj. classes ℭ of G such that |ℭ|_{p} = |G|_{p}} ,

where | . |_{p} denotes the p-part of the size (| . |_{0} = 1).

(2) If H is involutory then the implication (3) ⇒ (1) above follows more simply from Theorem 2.3(b) which asserts that, for H not semisimple, the map G_{0}(H) ⟶^{dim_{𝕜}} ℤ ↠ ℤ/pℤ factors through the cokernel of the Cartan map c. Now just use the fact that if det C ≠ 0 then det C is the size of this cokernel.

(3) If K is a local Hopf subalgebra of H (or, more generally, a local subalgebra so that H is free over K) then dim_{𝕜} K divides det C. Indeed, since G_{0}(K) = ⟨[𝕜_{ε_{K}}]⟩ and projective H-modules are projective, and hence free, over K, the restriction map induces an epimorphism

G_{0}(H)/cK_{0}(H) ↠^{Res} G_{0}(K)/cK_{0}(K) ≅ ℤ/ dim_{𝕜} K · ℤ .

3.5 Trace spaces for Hopf algebras. [R], Lemma 1 gives two alternative descriptions of the space of 𝕜-valued traces C(H). Namely, C(H) is the set of invariants with respect to the left coadjoint action of H on H^{∗} that is given by

(h ▷ f)(k) = ∑ f(S(h_{2})kh_{1}) for h, k ∈ H, f ∈ H^{∗}.

(This is the transpose of the action ◁ that was used in §3.4.) Alternatively, one can characterize C(H) as the set Cocom H^{∗} of cocommutative elements of H^{∗}. To summarize,

C(H) = Cocom H^{∗} = (H^{∗})^{H} .

Note that {f ∈ H^{∗} : f vanishes on J} is the coradical (H^{∗})_{0} of H^{∗} (cf. [Mo], Proposition 5.2.9). Therefore, C(H)_{reg} can be viewed as the set Cocom (H^{∗})_{0} of cocommutative elements of the coradical of H^{∗}. Since the coradical (H^{∗})_{0} is clearly mapped to itself under ▷, we have

C(H)_{reg} = Cocom (H^{∗})_{0} = (H^{∗})_{0}^{H} .

From the foregoing, we see in particular that C(H) is a 𝕜-subalgebra of H^{∗}, because Cocom H^{∗} clearly is. In Corollary 3.7 we will show that if H/J is separable then C(H)_{reg} is also a 𝕜-subalgebra of H^{∗} or, equivalently, [H, H] + J is a coideal of H. Note further that C(H) and C(H)_{reg} are both stable under the antipode S^{∗} of H^{∗}, since [H, H] and J = rad H are stable under S.

Examples. (1) H = (𝕜G)^{∗} for G a finite group: Here C(H)_{reg} = C(H) = H^{∗} = 𝕜G. Since 𝕜G need not be commutative, this example shows in particular that the isomorphism Z(H) ≅ C(H) of Lemma 1.7 is not in general an isomorphism of 𝕜-algebras.

(2) H = 𝕜G for G a finite group: Here, (h ▷ f)(g) = f(g^{h}) for g, h ∈ G and f ∈ H^{∗}, where g^{h} = h^{−1}gh is conjugation in G. Thus C(H) is canonically isomorphic with the 𝕜-algebra of all 𝕜valued functions on the set of conjugacy classes of G, with “pointwise” multiplication of functions. Furthermore, [H, H] + J is the 𝕜-linear span of the elements g − g_{p′}^{h} for g, h ∈ G, where g_{p′} denotes the p′-part of g (= g if p = 0). Therefore, T (H)_{reg} can be identified with the 𝕜-vector space with basis the set of p-regular conjugacy classes of G. Under this identification, the image [g] ∈ T (H)_{reg} of g ∈ G becomes the conjugacy class of p′-part of g. Furthermore, C(H)_{reg} can be viewed as the algebra of 𝕜-valued functions on the set of p-regular conjugacy classes of G: A trace f ∈ C(H) belongs to C(H)_{reg} precisely if it satisfies f(g) = f(g_{p′}) for all g ∈ G. See [P], p. 56–58 for all this.

3.6 Proposition. The character map χ : G_{0}(H) → C(H) is a ring homomorphism which satisfies χ_{V^{∗}} = S^{∗}(χ_{V}) and χ_{V} (1) = dim_{𝕜}(V). If 𝕜 is a splitting field for H then C(H)_{reg} is a 𝕜-subalgebra of H^{∗} and χ induces an isomorphism of 𝕜-algebras

χ̃ = Id_{𝕜} ⊗χ : { \atop G_{0}(H)} = 𝕜 ⊗_{ℤ} G_{0}(H) ⟶^{≅} C(H)_{reg} .

Proof. First, χ_{𝕜_{ε}} = ε, and so χ respects the identity elements of G_{0}(H) and C(H). Let V and W be H-modules. Then, by §2.1, there are 𝕜-isomorphisms End_{𝕜}(V ⊗ W) ≅ V ⊗ W ⊗ W^{∗} ⊗ V^{∗} ≅ End_{𝕜}(V) ⊗ End_{𝕜}(W). Under this identification, tr_{V ⊗W} becomes

tr_{V ⊗W} : End_{𝕜}(V ⊗ W) ≅ End_{𝕜}(V) ⊗ End_{𝕜}(W) ⟶^{tr_{V} ⊗ tr_{W}} 𝕜 ⊗ 𝕜 ≅ 𝕜 ,

and

h_{V ⊗W} = ∑(h_{1})_{V} ⊗ (h_{2})_{W} (h ∈ H) ,

where h ↦ h_{V} denotes the structure map H → End_{𝕜}(V), etc.. Therefore, the fact that χ is a ring homomorphism follows from the computation χ_{V ⊗W} (h) = ∑ tr_{V} (h_{1})_{V} tr_{W} (h_{2})_{W} = ∑ χ_{V} (h_{1})χ_{W} (h_{2}) = (χ_{V} χ_{W}) (h). The formula χ_{V^{∗}} = S^{∗}(χ_{V}) follows similarly, since the obvious 𝕜-linear isomorphism End_{𝕜}(V^{∗}) ≅ End_{𝕜}(V) sends h_{V^{∗}} ↦ S(h)_{V}. The formula χ_{V} (1) = dim_{𝕜}(V) is trivial. The remaining assertions now follow from the fact that, for 𝕜 a splitting field, the map χ̃ is an isomorphism between { \atop G_{0}(H)} and C(H)_{reg}, by Theorem 1.6(b). □

3.7 Corollary. If H/J is separable over 𝕜 then C(H)_{reg} is a 𝕜-subalgebra of H^{∗}.

Proof. Choose a field extension 𝕂/𝕜 so that 𝕂 is a splitting field for H̃ = 𝕂 ⊗ H. By Proposition 3.6, C(H̃)_{reg} is a 𝕂-subalgebra of H̃^{∗} = 𝕂 ⊗ H^{∗}. By separability, rad H̃ = 𝕂 ⊗ J and so C(H̃)_{reg} = 𝕂 ⊗ C(H)_{reg}. Therefore, C(H)_{reg} = C(H̃)_{reg} ∩ H^{∗} is closed under multiplication, since both H^{∗} and C(H̃)_{reg} are. □

Remark. Of course, H/J is separable in case the base field 𝕜 is perfect. In addition, H/J is known to be separable if either J = 0 (cf. [Mo], Cor. 2.2.2(1)), or if H = 𝕜G is the group algebra of a finite group G ([CR], Theorem 7.10), or if H = u(𝔤) is the restricted enveloping algebra of a “classical” p-Lie algebra 𝔤 ([Se], p. 101). I am not aware of an example where H/J is not separable.

4. Examples

4.1 The Sweedler algebra. Let H = H_{4} = 𝕜⟨1, g, x, gx| g^{2} = 1, x^{2} = 0, xg = −gx⟩ and assume that 𝕜 has characteristic p ≠ 2. This algebra is quasitriangular but not unimodular (cf. [Mo], 1.5.6, 2.1.2, 10.1.17). Here, J = xH and H/J ≅ 𝕜⟨g⟩ ≅ (𝕜⟨g⟩)^{∗}. Thus J is a Hopf ideal, and so we obtain ring isomorphisms G_{0}(H) ≅ ℤ⟨g⟩. The two irreducible H-modules are 𝕜_{ε} and 𝕜_{α}, where α : g ↦ −1, x ↦ 0 is the distinguished group-like element of H^{∗}. Their projective covers, P (𝕜_{ε}) and P (𝕜_{α}), are dual to each other, by Lemma 2.5. Therefore, P (𝕜_{ε}) has socle isomorphic to 𝕜_{α}, and the socle of P (𝕜_{α}) is isomorphic to 𝕜_{ε}. Thus, in G_{0}(H), we have c([P (𝕜_{ε})]) = c([P (𝕜_{α})]) = [𝕜_{ε}] + [𝕜_{α}], and so the Cartan matrix is

C = ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) .

Note that the foregoing applies verbatim whenever H has only two irreducible modules, 𝕜_{ε} and 𝕜_{α}, where α is the distinguished group-like element. A cocommutative example of this form is the restricted enveloping algebra H = u(𝔤) of the restricted 2-dimensional solvable 2-Lie algebra 𝔤 in characteristic p = 2 (cf. §4.3 below).

4.2 Radford’s algebra. The following Hopf algebra H is taken from [R3], Example 1 which in turn is based on an earlier construction due to Taft [T]. Fix n > 1 and assume that 𝕜 contains a root of unity ω of order n. Then H is generated by elements x, y, g subject to the relations

g^{n} = 1, x^{n} = y^{n} = 0, xg = ωgx, gy = ωyg, xy = ωyx .

Furthermore, g is group-like, and x and y are g, 1-primitive. The Hopf algebra has 𝕜-basis {g^{ℓ}x^{r}y^{s} : 0 ≤ ℓ, r, s < n}, H is unimodular, and S^{2}(h) = ghg^{−1} holds for all h ∈ H (see [R3]). Thus H is symmetric. The radical of H is J = xH + yH, a Hopf ideal of H, and H/J ≅ 𝕜⟨g⟩. The simple H-modules are: 𝕜_{ϕ^{i}} (i = 0, …, n − 1), where ϕ ∈ G(H^{∗}) is given by ϕ(g) = ω, ϕ(x) = ϕ(y) = 0, and the map [𝕜_{ϕ^{i}}] ↦ g^{i} gives a ring isomorphism G_{0}(H) → ℤ⟨g⟩. Putting e = \frac{1}{n} ∑ g^{i} ∈ H we have e = e^{2} and P (𝕜_{ε}) = He = ⨁_{0≤r,s<n} 𝕜x^{r}y^{s}e. The Loewy factors of P (𝕜_{ε}) are given by

J^{ℓ}P (𝕜_{ε})/J^{ℓ+1}P (𝕜_{ε}) = ∑_{r+s=ℓ} 𝕜x^{r}y^{s}e mod J^{ℓ+1}P (𝕜_{ε}) ≅ ⨁_{r+s=ℓ} 𝕜_{ϕ^{s−r}} .

Therefore, in G_{0}(H) = ℤ⟨g⟩, we have

c([P (𝕜_{ε})]) = ∑_{ℓ=0}^{2n−2} ∑_{r+s=ℓ}[𝕜_{ϕ^{s−r}}] = (1 + g^{−1} + … + g^{−(n−1)}) · (1 + g + … + g^{n−1}) = n(1 + g + … + g^{n−1}) .

Corollary 3.3 now implies that c([P (𝕜_{ϕ^{i}})]) = n(1 + g + … + g^{n−1})g^{i} = n(1 + g + … + g^{n−1}) holds for all i. Hence the Cartan matrix is

C = n · ( \begin{matrix}1 & & & 1 & & … & & & 1 \\ 1 & & & 1 & & … & & & 1 \\ . & . & . & . & . & . & . & . & . & . & . & . \\ 1 & & & 1 & & … & & & 1 \end{matrix} ) .

4.3 Restricted enveloping algebras of completely solvable p-Lie algebras. Let H = u(𝔤) be the restricted enveloping algebra of the finite dimensional completely solvable restricted p-Lie algebra (𝔤, [p]) over an algebraically closed field 𝕜 of characteristic p > 2. If the p-map [p] is normalized so as to be trivial on the center of 𝔤, then 𝔤 = 𝔱⊕𝔯, where 𝔯 = rad_{p}(𝔤) is the p-radical of 𝔤 and 𝔱 is a maximal torus. Furthermore, J = 𝔯H and H/J ≅ u(𝔱) is commutative. (See [SF], p. 240 ff and Theorem 5.3, p. 221 for all this.) In particular, by §3.1 Remarks and Examples (3), there is an isomorphism

G_{0}(H) ≅ ℤG , [𝕜_{α}] ↦ α (α ∈ G) ,

where G is the group of group-like elements of H^{∗}, written multiplicatively. Note that G is canonically isomorphic with {α ∈ 𝔤^{∗}| α([𝔤, 𝔤]) = 0, α(x^{[p]}) = α(x)^{p} ∀x ∈ 𝔤}, and hence G is an elementary abelian p-group of rank dim_{𝕜} 𝔱 (cf. [SF], Proposition 8.8 and Exercise 4, p. 243). So

|G| = p^{g} with g = dim_{𝕜} 𝔱 .

Putting T = u(𝔱) and R = u(𝔯), we have H ≅ R ⊗ T as 𝕜-vector spaces, by the restricted Poincaré-Birkhoff-Witt Theorem ([SF], Theorem 5.1). Choose an integral e ∈ T with ε(e) = 1. Then e is idempotent, and

P (𝕜_{ε}) = He = Re ≅ R

as H-modules, viewing R as H-module by means of the left adjoint representation ad_{ℓ} that is given by (ad_{ℓ}h)(r) = ∑ h_{1}rS(h_{2}). (The isomorphism follows from the calculation hre = ∑ h_{1}rε(h_{2})e = ∑ h_{1}rε(Sh_{2})e = ∑ h_{1}rS(h_{2})e.) In particular,

dim_{𝕜} P (𝕜_{ε}) = p^{r} with r = dim_{𝕜} 𝔯 .

In order to determine c([P (𝕜_{ε})]) ∈ G_{0}(H), it suffices to analyze R as a T -module. To this end, decompose 𝔯 as a T -module. This amounts to choosing a basis {x_{1}, …, x_{r}} of 𝔯 such that (ad_{ℓ}y)(x_{i}) = α_{i}(y)x_{i} holds for all y ∈ 𝔱, where α_{i} ∈ 𝔱^{∗} are suitable linear forms. We will view the α_{i} as elements of G by inflation and call them the Jordan-Hölder values of 𝔤 in 𝔯 (following [BGR]). By the PBW Theorem, we have T -module isomorphisms

R ≅ 𝕜[x_{1}] ⊗ 𝕜[x_{2}] ⊗ … ⊗ 𝕜[x_{r}] , where 𝕜[x_{i}] = ⨁_{j=0}^{p−1} 𝕜x_{i}^{j} ≅ ⨁_{j=0}^{p−1} 𝕜_{jα_{i}} .

Thus, as T -modules,

R ≅ ⨁_{j_{1},... ,j_{r}=0}^{p−1} 𝕜_{j_{1}α_{1}+...+j_{r}α_{r}} .

Viewing the α_{i} as elements of G as above, this isomorphism yields the following equation for [P (𝕜_{ε})] = [R] in G_{0}(H) = ℤG:

[P (𝕜_{ε})] = ∑_{j_{1},... ,j_{r}=0}^{p−1} α_{1}^{j_{1}} … α_{r}^{j_{r}} .

Let A ⊆ G denote the subgroup that is generated by the Jordan-Hölder values {α_{i}} and let a (≤ r) denote the rank of A. Then the above expression for [P (𝕜_{ε})] can be written as

[P (𝕜_{ε})] = p^{r−a}Â , where Â = ∑_{α∈A} α ∈ ℤG .

By Corollary 3.3, the projective covers of the remaining simple H-modules 𝕜_{α} (α ∈ G) are given by [P (𝕜_{α})] = [P (𝕜_{ε})]α. Consequently, the Cartan matrix C has the form

C = p^{r−a} · ( \begin{matrix}B \\ & . & . & . \\ & & & & B \end{matrix} )_{p^{g−a}×p^{g−a}} with B = ( \begin{matrix}1 & & & 1 & & … & & & 1 \\ 1 & & & 1 & & … & & & 1 \\ . & . & . & . & . & . & . & . & . & . & . & . \\ 1 & & & 1 & & … & & & 1 \end{matrix} )_{p^{a}×p^{a}} .

We briefly discuss some explicit examples.

(1) The 2-dimensional solvable p-Lie algebra. This Lie algebra is defined by 𝔤 = 𝕜t ⊕ 𝕜x with [t, x] = x, t^{[p]} = t, x^{[p]} = 0. Here, 𝔱 = 𝕜t, 𝔯 = 𝕜x, and G = A = ⟨α⟩ with α(t) = 1, α(x) = 0. This is also the only Jordan-Hölder value of 𝔤 in 𝔯. So g = r = a = 1 and we obtain

[P (𝕜_{ε})] = Ĝ ∈ ℤG and C = ( \begin{matrix}1 & & & 1 & & … & & & 1 \\ 1 & & & 1 & & … & & & 1 \\ . & . & . & . & . & . & . & . & . & . & . & . \\ 1 & & & 1 & & … & & & 1 \end{matrix} )_{p×p} .

(2) The diamond algebra. Here, 𝔤 = 𝕜t ⊕ 𝕜x ⊕ 𝕜y ⊕ 𝕜z with [t, x] = x, [t, y] = −y, [x, y] = z, [𝔤, z] = 0, and t^{[p]} = t, x^{[p]} = y^{[p]} = z^{[p]} = 0. Again, 𝔱 = 𝕜t, 𝔯 = 𝕜x ⊕ 𝕜y ⊕ 𝕜z, and G = A = ⟨α⟩ with α(t) = 1, α(x) = α(y) = α(z) = 0. The Jordan-Hölder values in 𝔯 are α, −α, and 0. Thus g = a = 1, r = 3, and so

[P (𝕜_{ε})] = p^{2}Ĝ ∈ ℤG and C = p^{2} · ( \begin{matrix}1 & & & 1 & & … & & & 1 \\ 1 & & & 1 & & … & & & 1 \\ . & . & . & . & . & . & . & . & . & . & . & . \\ 1 & & & 1 & & … & & & 1 \end{matrix} )_{p×p} .

4.4 The restricted enveloping algebra of sl(2, 𝕜) (cf. [Po]). We now consider the restricted enveloping algebra H = u(𝔤) of the p-Lie algebra 𝔤 = sl(2, 𝕜) over a field 𝕜 of characteristic p > 2. Thus 𝔤 = 𝕜f ⊕ 𝕜h ⊕ 𝕜e, with [h, f] = −2f, [h, e] = 2e, [e, f] = h, and the p-map is given by e^{[p]} = f^{[p]} = 0, h^{[p]} = h. Let V = 𝕜a ⊕ 𝕜b denote the canonical 𝔤-module that is given by the matrices

h_{V} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , e_{V} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , f_{V} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) .

The symmetric group S_{m} acts on the m-fold tensor product V^{⊗m} by permuting positions, and this action commutes with the H-action on V^{⊗m}, since H is cocommutative. Thus the S_{m}-fixed points form an H-submodule of V^{⊗m} which we denote by V (m):

V (m) = (V^{⊗m})^{S_{m}} .

By [Bou], Proposition 4 on p. IV.44,

dim_{𝕜} V (m) = m + 1 .

The modules V (m) (0 ≤ m < p) form a complete set of irreducible H-modules, and they are in fact absolutely irreducible (cf. [C] or [SF], p. 207ff). Furthermore, all V (m) (0 ≤ m < p) are self-dual, by uniqueness of dimension. Note that V (0) = 𝕜_{ε} and V (1) = V. In the following, we will write v_{m} = [V (m)] ∈ G_{0}(H) and v = [V] ∈ G_{0}(H). Thus v_{0} = 1 and

G_{0}(H) = ⨁_{m=0}^{p−1} ℤv_{m} ≅ ℤ^{p} .

The ring structure of G_{0}(H). Identifying S_{m−1} with the subgroup Stab_{S_{m}}(m) ⊆ S_{m} we have (V^{⊗m})^{S_{m−1}} = V (m−1)⊗V which shows that V (m) is a submodule of V (m−1)⊗V for all m. Furthermore, if m ≥ 2 then V (m−1)⊗V contains the element x = (m−1)!a^{⊗m−1}⊗b−∑_{σ∈S_{m−1}} (b ⊗ a^{⊗m−1})^{σ} which satisfies hx = (m−2)x, ex = 0, and f^{m−1}x = 0. It follows that, for 2 ≤ m < p+2, Hx is the simple module V (m−2). For m < p, the submodules V (m−2) and V (m) of V (m−1)⊗V intersect trivially, and so V (m−1)⊗V = V (m)⊕V (m−2) if 2 ≤ m < p, by counting dimensions. For m = p, the element y = a^{⊗p} ∈ V (p) satisfies hy = py = 0 = ey, and the element x belongs to V (p), since (p − 1)! = −1 mod p. Therefore, V (p) has V (p−2) and V (0) as composition factors, and there must be another copy of V (0), for dimension reasons. Since V (p) ⊆ V (p−1)⊗V, we conclude that V (p−1)⊗V has 2 copies of V (0) and at least one copy of V (p−2) as composition factors. There actually is a second copy of V (p−2). For, Hom_{H}(V (p−1)⊗V, V (p−2)) ≅ Hom_{H}(V (p−1), V (p−2)⊗V) = Hom_{H}(V (p−1), V (p−1)⊕V (p−3)) shows that there is an epimorphism π : V (p − 1) ⊗ V ↠ V (p − 2), and π does not split, because V (p − 1) ⊗ V is projective while V (p − 2) is not (see below).

Summarizing, we have shown that the following equations hold in G_{0}(H):

v_{0} · v = v_{1}, v_{m−1} · v = v_{m} + v_{m−2} (2 ≤ m < p) and v_{p−1} · v = 2v_{p−2} + 2v_{0}.

It follows easily by induction that ∑_{i=0}^{m} ℤv_{i} = ∑_{i=0}^{m} ℤv^{i} holds for all m = 0, …, p − 1. In particular,

G_{0}(H) = ℤ[v] ≅ ℤ[X]/(g(X)) ,

where g(X) is the characteristic polynomial of the p × p-matrix of the endomorphism of G_{0}(H) = ℤ^{p} that is given by multiplication with v. One can show that

g(X) = (2 − X)f_{p}(X)^{2} ,

where

f_{p}(X) = ∑_{k=0}^{(p−1)/2}(−1)^{a_{p,k}} \binom{k + a_{p,k}}{a_{p,k}}X^{k}, a_{p,k} = ⌊\frac{1}{4}(p − 1 − 2k)⌋ .

Consequently, G_{0}(H) is not a semiprime ring. We also remark that f_{p}(X) is irreducible and satisfies f_{p}(X) = (X − 2)^{\frac{p−1}{2}} mod p. In particular:

C(H)_{reg} ≅ k[X]/ ((X − 2)^{p}) ≅ kC_{p} ,

where C_{p} is the cyclic group of order p.

The principal indecomposable modules. We now discuss the projective covers

P (m) = P (V (m)) (0 ≤ m < p).

By Theorem 2.3(b), we know that dim_{𝕜} P (m) = pℓ_{m} for some ℓ_{m} ∈ ℕ. In particular, if m < p − 1 then V (m) is not projective, and consequently P (m) has at least 2 constituents V (m), one in the head and another one in the socle. (Recall that H is a symmetric algebra.) This forces dim_{𝕜} P (m) ≥ 2(m + 1). Thus, if \frac{p−1}{2} ≤ m < p−1 then we must have ℓ_{m} ≥ 2. Furthermore, using the isomorphism V (m)⊗V ≅ V (m + 1) ⊕ V (m − 1) for 1 ≤ m < p − 1, we see that the projective module P (m) ⊗ V maps onto ≅ V (m + 1) ⊕ V (m − 1), and hence P (m + 1) ⊕ P (m − 1) is a direct summand of P (m) ⊗ V. This implies ℓ_{m+1} + 1 ≤ 2ℓ_{m}, and it follows by induction that ℓ_{m} ≥ 2 holds for all m < p − 1. On the other hand, since 𝕜 is a splitting field for H, we have an isomorphism

H ≅ ⨁_{m=0}^{p−1} P (m)^{m+1} .

Counting dimensions, we conclude that ℓ_{m} = 2 for m < p − 1 and ℓ_{p−1} = 1. Thus

dim_{𝕜} P (m) = 2p (0 ≤ m < p − 1) and P (p − 1) ≅ V (p − 1) .

In particular, V (p − 1) is the only projective module of dimension p. Since the projective module V (p − 1) ⊗ V maps onto V (p − 2), we have

P (p − 2) ≅ V (p − 1) ⊗ V .

Using the fact that P (m + 1) ⊕ P (m − 1) is a direct summand of P (m) ⊗ V for 1 ≤ m < p − 1, we obtain the following isomorphisms

P (m) ⊗ V ≅ P (m + 1) ⊕ P (m − 1) (1 ≤ m < p − 2) and P (p − 2) ⊗ V ≅ P (p − 3) ⊕ V (p − 1)^{2} .

Put a_{m} = [P (m)] ∈ K_{0}(H), so a_{0}, a_{1}, …, a_{p−1} is the canonical ℤ-basis of K_{0}(H). An easy induction shows that a_{p−1} = [V (p − 1)] generates K_{0}(H) as G_{0}(H)-module, and so K_{0}(H) is free over G_{0}(H).

The Cartan map. We claim that the Cartan map c : K_{0}(H) → G_{0}(H) is given by

c(a_{p−1}) = v_{p−1} and c(a_{m}) = 2v_{m} + 2v_{p−2−m} (0 ≤ m < p − 1).

Indeed, the first formula is clear from P (p−1) ≅ V (p−1), and the second formula for m = p−2 follows from P (p − 2) ≅ V (p − 1) ⊗ V which implies c(a_{p−2}) = v_{p−1}v = 2v_{p−2} + 2v_{0}. For m = p − 3, we use the isomorphism P (p − 2) ⊗ V ≅ P (p − 3) ⊕ V (p − 1)^{2} to obtain c(a_{p−3}) = c(a_{p−2})v − 2v_{p−1} = (2v_{p−2} + 2v_{0})v−2v_{p−1} = 2(v_{p−1} +v_{p−3})+2v−2v_{p−1} = 2v_{p−3} +2v_{1}, as required. The assertion for the remaining 0 ≤ m < p − 3 now follows by induction, based on the isomorphism P (m + 1) ⊗ V ≅ P (m + 2) ⊕ P (m), or c(a_{m}) = c(a_{m+1})v − c(a_{m+2}). Therefore, the Cartan matrix has the form

C = ( \begin{matrix}2 & 0 & … & 0 & 2 & 0 \\ 0 & 2 & … & 2 & 0 & 0 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & 2 & … & 2 & 0 & 0 \\ 2 & 0 & … & 0 & 2 & 0 \\ 0 & & & . . . . . . . . . . . . & & & 1 \end{matrix} )_{p×p} .

Loewy series. The Loewy series of the principal indecomposable modules P (m) (0 ≤ m < p − 1) follows easily. Indeed, the head P (m)/P (m)J and the socle ann_{P (m)}(J) are both isomorphic to V (m), and the core P (m)J/socle P (m) consists of two copies of V (p − 2 − m). It follows that Ext_{H}(V (m), V (m)) = 0 for 0 ≤ m < p − 1 (and of course also for the projective module V (p − 1)), and hence the core of P (m) must actually be isomorphic with V (p − 2 − m) ⊕ V (p − 2 − m). Thus, for 0 ≤ m < p − 1, we have

P (m) = \begin{matrix}V (m) \\ V (p − 2 − m) ⊕ V (p − 2 − m) \\ V (m) \end{matrix} .

Consequently, the radical J = rad H has nilpotence index 3. This could of course also be checked using the explicit generators e^{p−1}(h + 1) and (h + 1)f^{p−1} for J that are given in [Se], p. 99.

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