ON BLOCKS OF DEFECT TWO AND ONE SIMPLE MODULE, AND LIE ALGEBRA STRUCTURE OF HH^{1}

D. J. BENSON, RADHA KESSAR, AND MARKUS LINCKELMANN

Abstract. Let k be a field of odd prime characteristic p. We calculate the Lie algebra structure of the first Hochschild cohomology of a class of quantum complete intersections over k. As a consequence, we prove that if B is a defect 2-block of a finite group algebra kG whose Brauer correspondent C has a unique isomorphism class of simple modules, then a basic algebra of B is a local algebra which can be generated by at most 2\sqrt{I} elements, where I is the inertial index of B, and where we assume that k is a splitting field for B and C.

1. Introduction

The purpose of this paper is to examine certain algebras of dimension p^{2} over a field of odd characteristic p, which occur as the basic algebras of blocks of finite groups with normal defect groups of order p^{2} and a unique simple module. The goal is to understand the Brauer correspondents of such blocks. To this end, we make a detailed examination of the degree one Hochschild cohomology as a Lie algebra.

Theorem 1.1. Let k be a field of odd prime characteristic p and let q ∈ k^{×} be an element of finite order e such that e ≥ 2 and such that e divides p − 1. Let

A = k⟨x, y | x^{p} = 0 = y^{p}, yx = qxy⟩ .

Set L = HH^{1}(A) and let L′ be the derived Lie subalgebra of the Lie algebra L. Denote by soc_{Z(A)}(L) the socle of L as a left Z(A)-module. Then A is a split local symmetric k-algebra of dimension p^{2}, and the following hold.

(i) We have dim_{k}(L) = 2(p + (\frac{p−1}{e})^{2}).

(ii) We have Z(L) = {0}.

(iii) There is a 2-dimensional maximal toral subalgebra H of L such that L = H ⊕ L′. (iv) The derived subalgebra L′ is nilpotent; in particular, L is solvable.

(v) We have dim_{k}(soc_{Z(A)}(L)) = 2e and soc_{Z(A)}(L) ⊆ Z(L′).

(vi) We have J(Z(A))L = L′ and dim_{k}(L/L′) = 2.

(vii) We have dim_{k}(Z(L′)) = 2e + 2. In particular, L′ is abelian if and only if e = p − 1. (viii) The subalgebra H is p-toral, and we have (L′)^{[p]} = {0}.

See Section 5 for the proof. Other papers examining Hochschild cohomology of similar algebras include Bergh and Erdmann [1] and Oppermann [15], but their results and goals lie in different directions. For example, in [1] it is assumed that q is not a root of unity. The last statement in Theorem 1.1 regarding the p-restricted structure of L is motivated by invariance results of p-power maps in Hochschild cohomology under derived and stable equivalences in work of Zimmermann [23] and Rubio y Degrassi [18].

Date: April 15, 2016.

To exploit Theorem 1.1 we prove the following general theorem, which provides an upper bound for the number of loops in the quiver of a symmetric split algebra over an arbitrary field.

Theorem 1.2. Let k be a field and let A be a symmetric split k-algebra. We have

∑_{S} dim_{k}(Ext_{A}^{1}(S, S)) ≤ dim_{k}(soc_{Z(A)}(HH^{1}(A)))

where in the sum S runs over a set of representatives of the isomorphism classes of simple A-modules. In particular, if A is a symmetric split local k-algebra, then

dim_{k}(J(A)/J(A)^{2}) ≤ dim_{k}(soc_{Z(A)}(HH^{1}(A))) .

This will be proved in Theorem 3.1 and Corollary 3.2. Combining the two theorems above with standard properties of stable equivalences of Morita type yields the following consequence.

Corollary 1.3. Let A be as in Theorem 1.1, and let B be a split local symmetric k-algebra such that there is a stable equivalence of Morita type between A and B. We have

dim_{k}(J(B)/J(B)^{2}) ≤ 2e .

The motivation for the above results comes from local-global considerations in the modular representation theory of finite groups. Let G be a finite group and let B be a block of the group algebra kG of G over a field k of odd characteristic. Let P be a defect group of B, C the block of of kN_{G}(P) in Brauer correspondence with B and let I be the inertial index of B. Suppose that k is a splitting field for B and C. If P has order p^{2}, then it is known that there is a stable equivalence of Morita type between B and C. If in addition C has a unique isomorphism class of simple modules, then C is a matrix algebra over a quantum complete intersection as in Corollary 1.3. Moreover, in this case e ≤ \sqrt{I} and if I > 1, then e > 1. Thus, Corollary 1.3 yields the following local-global result.

Corollary 1.4. Let G be a finite group and let B be a block of the group algebra kG of G over a field k of odd characteristic p. Let P be defect group of B, C the block of kN_{G}(P) in Brauer correspondence with B and let I be the inertial index of B. Suppose that P has order p^{2}, that C has a unique isomorphism class of simple modules, and that k is a splitting field for B and C. Then B has a unique isomorphism class of simple modules, and

dim_{k}(J(B)/J(B)^{2}) ≤ 2\sqrt{I} .

Corollaries 1.3 and 1.4 are proved at the end of Section 5. We note that in the situation of Corollary 1.4, Broué’s abelian defect group conjecture [3] would imply that the blocks B and C are derived equivalent, and therefore by a result of Roggenkamp and Zimmermann [22, Proposition 6.7.4], that B and C are Morita equivalent. Hence, it would follow that the dimension of J(B)/J(B)^{2} is two. If p = 3, it is known that B and C are Morita equivalent in this situation [8].

If e = 2, then the algebra A in Theorem 1.1 is Morita equivalent to the nonprincipal block algebra of the finite group algebra kG, where G = (C_{p}×C_{p})⋊Q_{8}, with Z(Q_{8}) acting trivially on C_{p} × C_{p}, such that the induced action of Q_{8}/Z(Q_{8}) ≅ C_{2} × C_{2} is given by each copy of C_{2} acting by inversion on the corresponding copy of C_{p}. Thus A lifts to an O-free O-algebra Â which is Morita equivalent to the nonprincipal block B_{1} of OG. Here O is a complete discrete valuation ring of characteristic zero with residue field k of odd prime characteristic p; we assume that O contains a primitive 4p-th root of unity. This algebra Â can be described, using the normalised polynomials f_{n}(u) = 2T_{n}(\frac{u}{2}) of the Chebyshev polynomials of the first kind T_{n} (see §6 for a more detailed review of the notation).

Theorem 1.5. With the notation above, the O-algebra

Â = O⟨γ, δ | γδ + δγ = 0, f_{p}(γ) = 0 = f_{p}(δ)⟩

is a basic algebra of B_{1}. In particular, we have k ⊗_{O} Â ≅ A.

This will be proved in §6. If e > 2, it turns out that it is much harder to describe Â.

2. Basic background facts

Let k be a field. For A a finite-dimensional k-algebra, we denote by ℓ(A) the number of isomorphism classes of simple A-modules. We write A^{e} = A⊗_{k} A^{op}. We consider A^{e}-modules as A-A-bimodules and vice versa, whenever convenient. We denote by [A, A] the additive commutator space, spanned by the set of elements [a, b] = ab − ba, with a, b ∈ A. If A is split local, then every element in A is of the form λ · 1 + r for some λ ∈ k and some r ∈ J(A). This yields immediately the following well-known fact:

Lemma 2.1. Let A be a finite-dimensional split local k-algebra. We have [A, A] ⊆ J(A)^{2}.

A k-algebra A is symmetric if A is isomorphic to its k-dual A^{∨} as an A-A-bimodule (this implies that A is finite-dimensional). If A is symmetric, then the socle of A as a left Amodule and as a right A-module coincide. If A is also split, then this coincides with the socle of A as an A-A-bimodule. The image s ∈ A^{∨} of 1_{A} ∈ A under an A-A-bimodule isomorphism A ≅ A^{∨} is called a symmetrising form. Note that it satisfies s(ab) = s(ba). If A is symmetric with a fixed choice of a symmetrising form s, for any subspace U of A we denote by U^{⊥} the subspace consisting of all a ∈ A satisfying s(au) = 0 for all u ∈ U. We have dim_{k}(U) + dim_{k}(U^{⊥}) = dim_{k}(A), and hence U^{⊥⊥} = U. It is well-known that [A, A]^{⊥} = Z(A) and that soc(A)^{⊥} = J(A). The space [A, A] is contained in any symmetrising form of A. If A is split local symmetric, then soc(A) has dimension 1 and is the unique minimal ideal in A; thus, in that case, we have [A, A] ∩ soc(A) = {0}.

Lemma 2.2. Let A be a split local symmetric k-algebra. Then soc^{2}(A) ⊆ Z(A).

Proof. Choose a symmetrising form of A. The statement follows from Lemma 2.1, since (J(A)^{2})^{⊥} = soc^{2}(A) and [A, A]^{⊥} = Z(A). □

For A a split finite-dimensional k-algebra, the semisimple quotient A/J(A) is a direct product of matrix algebras, hence symmetric. Thus (A/J(A))^{∨} ≅ A/J(A) as A-A-bimodules. Moreover, we have an A-A-bimodule isomorphism A/J(A) ≅ ⊕_{S} S ⊗_{k} S^{∨}, where S runs over a set of representatives of the isomorphism classes of simple A-modules. If A is split and symmetric, then A/J(A) ≅ soc(A) and (A/soc(A))^{∨} ≅ J(A) as A-A-bimodules.

Lemma 2.3 ([5, Chapter IX, Corollary 4.4]). Let A be a finite-dimensional k-algebra, and let S, T be finite-dimensional A-modules. There is a canonical graded k-linear isomorphism

HH^{∗}(A; S ⊗_{k} T^{∨}) ≅ Ext_{A}^{∗}(T, S).

Proof. A standard adjunction, with T viewed as an A-k-bimodule, yields for any projective A^{e}-module P a natural isomorphism

Hom_{A}(P ⊗ T, S) ≅ Hom_{A^{e}}(P, Hom_{k}(T, S)) ≅ Hom_{A^{e}}(P, S ⊗_{k} T^{∨}).

By naturality, replacing P by a projective resolution of A as an A^{e}-module yields an isomorphism of cochain complexes. Taking cohomology yields the statement. □

Lemma 2.4. Let A be a split symmetric k-algebra. We have a graded k-linear isomorphism

HH^{∗}(A; soc(A)) ≅ ⊕_{S} Ext_{A}^{∗}(S, S) ,

where S runs over a set of representatives of the isomorphism classes of simple A-modules. In particular, we have

dim_{k}(Hom_{A^{e}}(A, soc(A))) = ℓ(A) ,
dim_{k}(HH^{1}(A; soc(A))) = ∑_{S} dim_{k}(Ext_{A}^{1}(S, S)) ,

where S runs over a set of representatives of the isomorphism classes of simple A-modules.

Proof. As mentioned above, we have A-A-bimodule isomorphisms

soc(A) ≅ A/J(A) ≅ ⊕_{S} S ⊗_{k} S^{∨},

where S runs over a set of representatives of the isomorphism classes of simple A-modules. Thus the isomorphism follows from the previous lemma. Comparing dimensions in degree 0 and in degree 1 yields the two equalities. □

3. Calculating derivations on symmetric algebras

Let k be a field and let A be a finite-dimensional k-algebra. We will use the description of HH^{1}(A) as outer derivations. A k-linear map f: A → A is a derivation if f(ab) = af(b) + f(a)b for all a, b ∈ A. If z ∈ Z(A) and f is a derivation on A, then z · f defined by (z · f)(a) = zf(a) is a derivation on A. In this way, the set of derivations Der(A) on A becomes a Z(A)-module. If x ∈ A, then the map [x, −] sending a ∈ A to [x, a] = xa − ax is a derivation; any derivation of this form is called an inner derivation, of A, and the set IDer(A) of inner derivations of A is a Z(A)-submodule of Der(A). We have a canonical isomorphism HH^{1}(A) ≅ Der(A)/IDer(A); see e.g. [21, 9.2.1]. The HH^{0}(A)-module structure and the Z(A)-module structure on Der(A)/IDer(A) correspond to each other through the canonical isomorphism HH^{0}(A) ≅ Z(A). Any derivation f on A satisfies f(1) = 0, since f(1) = f(1 · 1) = f(1) · 1 + 1 · f(1) = 2f(1), hence ker(f) is a unitary subalgebra of A. The space IDer(A) is isomorphic to the quotient of A by the kernel of the map x ↦ [x, −], hence dim_{k}(IDer(A)) = dim_{k}(A) − dim_{k}(Z(A)). Thus if A is symmetric, then dim_{k}(IDer(A)) = dim_{k}([A, A]). For any Z(A)-module H we denote by soc_{Z(A)}(H) its socle as a Z(A)-module.

Theorem 3.1. Let A be a symmetric split k-algebra and let E be a maximal semisimple subalgebra. Let f: A → A be an E-E-bimodule homomorphism satisfying E + J(A)^{2} ⊆ ker(f) and Im(f) ⊆ soc(A). Then f is a derivation on A in soc_{Z(A)}(Der(A)), and if f ≠ 0, then f is an outer derivation of A. In particular, we have

∑_{S} dim_{k}(Ext_{A}^{1}(S, S)) ≤ dim_{k}(soc_{Z(A)}(HH^{1}(A)))

where in the sum S runs over a set of representatives of the isomorphism classes of simple A-modules.

Proof. Let a, b ∈ A. By the Wedderburn–Malcev theorem, we have A = E ⊕ J(A). Thus a = c + r and b = d + s for some c, d ∈ E and r, s ∈ J(A). The hypotheses on f imply that

f(ab) = f(cd + cs + rd + rs) = f(cs + rd) = cf(s) + f(r)d = af(b) + f(a)b.

This shows that f is a derivation. Suppose that f is an inner derivation. Then Im(f) ⊆ [A, A]∩soc(A). But Im(f) is also an E-E-bimodule. Any E-E-bimodule contained in soc(A) is in fact an ideal. The space [A, A] contains no nonzero ideal as A is symmetric. Thus f is either zero or an outer derivation. Since J(Z(A)) is contained in J(A), any such derivation is annihilated by J(Z(A)). This shows that Hom_{E^{e}}(J(A)/J(A)^{2}, soc(A)) is isomorphic to a subspace of soc_{Z(A)}(HH^{1}(A)). Since J(A) annihilates J(A)/J(A)^{2} and soc(A), this subspace is isomorphic to Hom_{A^{e}}(J(A)/J(A)^{2}, soc(A)). As A is symmetric, we have

soc(A) ≅ A/J(A) ≅ ⊕_{S} S ⊗_{k} S^{∨},

with S running over a set of representatives of the isomorphism classes of simple A-modules. The dimension of Hom_{A^{e}}(J(A)/J(A)^{2}, S ⊗_{k} S^{∨}) is equal to the number of summands of the A^{e}-module J(A)/J(A)^{2} isomorphic to S ⊗_{k} S^{∨}. If i is a primitive idempotent such that iS ≠ {0}, then S is the unique simple quotient of Ai, hence S^{∨} is the unique simple quotient of iA, and thus S^{∨}i is one-dimensional. It then follows that the dimension of Hom_{A^{e}}(J(A)/J(A)^{2}, S ⊗_{k} S^{∨}) is equal to the number of summands of J(A)i/J(A)^{2}i isomorphic to S, and that is precisely dim_{k}(Ext_{A}^{1}(S, S)). □

Corollary 3.2. Let A be a split local symmetric k-algebra. Let f: A → A be a k-linear map satisfying 1 + J(A)^{2} ⊆ ker(f) and Im(f) ⊆ soc(A). Then f is a derivation on A in soc_{Z(A)}(Der(A)), and if f ≠ 0, then f is an outer derivation of A. In particular, we have

dim_{k}(J(A)/J(A)^{2}) ≤ dim_{k}(soc_{Z(A)}(HH^{1}(A))) .

Proof. Since A is split local, we have dim_{k}(J(A)/J(A)^{2}) = dim_{k}(Ext_{A}^{1}(S, S)), where S = A/J(A) is the unique simple A-module, up to isomorphism. Moreover, k · 1_{k} is the unique maximal semisimple subalgebra of A. The result follows from Theorem 3.1. □

Combining Theorem 3.1 and Corollary 3.2 implies Theorem 1.2.

Remark 3.3. Note that HH^{1}(A) is annihilated by the projective ideal Z^{pr}(A) in Z(A), hence HH^{1}(A) is a module over the stable center Z̄(A) = Z(A)/Z^{pr}(A), and

soc_{Z(A)}(HH^{1}(A)) = soc_{Z̄(A)}(HH^{1}(A)).

This shows that soc_{Z(A)}(HH^{1}(A)) is invariant under stable equivalences of Morita type.

It is possible to give a more structural proof of the inequality in Theorem 3.1, based on the following result.

Proposition 3.4. Let A be a split symmetric k-algebra. We have canonical short exact sequences

\xymatrix{0 \ar[r] & Hom_{A^{e}}(A, soc(A)) \ar[r] & Hom_{A^{e}}(A, A) \ar[r] & Hom_{A^{e}}(A, A/soc(A)) \ar[r] & 0}
\xymatrix{0 \ar[r] & Hom_{A^{e}}(A/J(A), A) \ar[r] & Hom_{A^{e}}(A, A) \ar[r] & Hom_{A^{e}}(J(A), A) \ar[r] & 0}

In particular, we have

dim_{k}(Z(A)) − ℓ(A) = dim_{k}(Hom_{A^{e}}(A, A/soc(A))) = dim_{k}(Hom_{A^{e}}(J(A), A)) .

Proof. We may assume that A is basic. Any A^{e}-homomorphism from A to A/soc(A) is in particular a homomorphism of left A-modules. As such, it lifts to an endomorphism of A, and hence is induced by right multiplication with an element y ∈ A, followed by the canonical map A → A/soc(A). For this to induce a bimodule homomorphism from A to A/soc(A) a necessary condition is [y, a] ∈ soc(A) for all a ∈ A. Using that A is basic, one can show that this forces y ∈ Z(A). Indeed, for any primitive idempotent i we have

[y, i] = yi − iy = yi − iyi − iy(1 − i) ∈ soc(A).

Since A is basic, this forces iy(1 − i) = 0 because soc(A(1 − i)) has no submodule isomorphic to the simple module Ai/J(A)i. Thus iy = iyi, and a similar argument shows iyi = yi. Thus y commutes with all primitive idempotents. For a ∈ Ai we have [y, a] = yai − ayi ∈ soc(Ai), so this is annihilated by 1 − i, hence equal to iyiai − iaiyi ∈ soc(iAi) ∩ [iAi, iAi], which is zero because the local algebra iAi is symmetric. Thus y ∈ Z(A), which means precisely that the induced homomorphism A → A/soc(A) lifts to a bimodule homomorphism A → A, whence the exactness of the first sequence as stated. The second sequence is obtained from applying duality to the first. By Lemma 2.4, the dimension of the left term in the first sequence is ℓ(A), and the middle term is isomorphic to Z(A), which proves the first equality. The second equality is obtained via duality. □

Remark 3.5. The inequality in Theorem 3.1 can be proved using Proposition 3.4 as follows. We consider the long exact sequence obtained from applying the functor Hom_{A^{e}}(A, −) to the short exact sequence of A^{e}-modules

\xymatrix{0 \ar[r] & soc(A) \ar[r] & A \ar[r] & A/soc(A) \ar[r] & 0}

This yields in particular an exact sequence

\xymatrix{Hom_{A^{e}}(A, A) \ar[r] & Hom_{A^{e}}(A, A/soc(A)) \ar[r] & HH^{1}(A; soc(A)) \ar[r] & HH^{1}(A)}

By Proposition 3.4 the first map is surjective. Thus the second map is zero, hence the third map is injective. Thus HH^{1}(A; soc(A)) is isomorphic to a subspace of HH^{1}(A). Since J(Z(A)) ⊆ J(A), this subspace is contained in soc_{Z(A)}(HH^{1}(A)). The inequality in Theorem 3.1 follows from 2.4.

The surjectivity of the first map in the above exact sequence can be used to give a proof of a result of Brandt [2], as follows. Identify Hom_{A^{e}}(J(A)/J(A)^{2}; A) with a subspace of Hom_{A^{e}}(J(A); A) via the canonical surjection J(A) → J(A)/J(A)^{2}. If J(A)^{2} is nonzero, then Hom_{A}(J(A)/J(A)^{2}, A) is strictly smaller than Hom_{A}(J(A), A), because the inclusion map J(A) ⊆ A does not factor through J(A)/J(A)^{2}. Since J(A)/J(A)^{2} is semisimple, we have

Hom_{A^{e}}(J(A)/J(A)^{2}, A) = Hom_{A^{e}}(J(A)/J(A)^{2}, soc(A)),

which in turn (as observed in the proof of 3.1) is isomorphic to ⊕_{S} Ext_{A}^{1}(S, S), where S runs over a set of representatives of the isomorphism classes of simple A-modules. Thus, if A is split symmetric such that J(A)^{2} ≠ {0}, then the dimension of Hom_{A^{e}}(J(A), A) is strictly greater than that of ⊕_{S} Ext_{A}^{1}(S, S). Proposition 3.4 implies in that case the inequality

dim_{k}(Z(A)) − ℓ(A) ≥ 1 + ∑_{S} dim_{k}(Ext_{A}^{1}(S, S))

due to Brandt [2, Theorem B].

It follows that the integers dim_{k}(Z(A)) − ℓ(A) − 1 and dim_{k}(soc_{Z(A)}(HH^{1}(A))) are both upper bounds for ∑_{S} dim_{k}(Ext_{A}^{1}(S, S)). These two upper bounds are not comparable in general, since they arise from unrelated parts of a long exact sequence with a zero map. The following two examples illustrate this. Suppose that k has odd prime characteristic p. If A = k(C_{p} ⋊ C_{p−1}), with C_{p−1} acting regularly on the nontrivial elements of C_{p}, then standard calculations yield

dim_{k}(Z(A)) − ℓ(A) − 1 = 0 < dim_{k}(soc_{Z(A)}(HH^{1}(A))) = 1 .

By contrast, if A is as in Theorem 1.1, then, using Lemma 4.3 below, we have

dim_{k}(Z(A)) − ℓ(A) − 1 = (\frac{p−1}{e})^{2} + 2p − 3 ≥ dim_{k}(soc_{Z(A)}(HH^{1}(A))) = 2e ,

with equality if and only if e = p − 1.

Derivations with image in the second socle layer are characterised as follows.

Proposition 3.6. Let A be a split local symmetric k-algebra, let {x_{1}, x_{2}, .., x_{r}} be a k-basis of a complement of J(A)^{2} in J(A), and let z be a nonzero element in soc(A). There is a basis {y_{1}, y_{2}, .., y_{r}} of a complement of soc(A) in soc^{2}(A) such that x_{i}y_{i} = y_{i}x_{i} = z for 1 ≤ i ≤ r, and such that x_{i}y_{j} = y_{j}x_{i} = 0, for 1 ≤ i, j ≤ r, i ≠ j. Let f: A → A be a k-linear map satisfying 1 + J(A)^{2} ⊆ ker(f), such that

f(x_{i}) = ∑_{j=1}^{r} σ_{i,j}y_{j}

for some coefficients σ_{i,j} ∈ k, 1 ≤ i, j ≤ r.

(i) The map f is a derivation if and only if σ_{i,j} = −σ_{j,i} for all i, j, 1 ≤ i, j ≤ r. In particular, if char(k) ≠ 2 and if f is a derivation, then σ_{i,i} = 0 for 1 ≤ i ≤ r, and the space of derivations obtained in this way has dimension \frac{r(r−1)}{2}.

(ii) If f is an inner derivation, then Im(f) ⊆ soc(Z(A)) ∩ soc^{2}(A) ∩ [A, A], and Im(f) is contained in a complement of soc(A) in soc(Z(A)) ∩ soc^{2}(A).

Proof. Let a, b ∈ A. In the following sums, the indices i and j run from 1 to r. Write

a = ∑_{i} α_{i}x_{i} + λ · 1 + u

with coefficients α_{i} and λ in k, and u ∈ J(A)^{2}. Similarly, write

b = ∑_{j} β_{i}x_{i} + μ · 1 + v

with β_{i}, μ ∈ k and v ∈ J(A)^{2}. Thus

f(a) = ∑_{i} α_{i}x_{i} = ∑_{i,j} α_{i}σ_{i,j}y_{j},
f(b) = ∑_{i} β_{i}x_{i} = ∑_{i,j} β_{i}σ_{i,j}y_{j}.

Since u, v annihilate the y_{i}, short calculations, using the hypotheses on f, yield

f(a)b = (∑_{i,j} α_{i}σ_{i,j}β_{j})z + μf(a),
af(b) = (∑_{i,j} β_{i}σ_{i,j}α_{j})z + λf(b),
f(ab) = μf(a) + λf(b).

Thus f is a derivation if and only if

∑_{i,j}(α_{i}σ_{i,j}β_{j} + β_{i}σ_{i,j}α_{j}) = 0

for all choices of coefficients α_{i}, β_{j}. This holds if and only if σ_{i,j} = −σ_{j,i} for all i, j. Statement (i) follows. Suppose now that f is an inner derivation, say f = [w, −] for some w ∈ A. By the assumptions on f, we have [w, J(A)^{2}] = {0} and [w, A] ⊆ soc^{2}(A) ⊆ Z(A), where the last inclusion is from Lemma 2.2. Note that Im(f) is spanned by the [w, x_{i}], 1 ≤ i ≤ r. If c ∈ J(Z(A)), then

[w, x_{i}]c = wx_{i}c − x_{i}wc = w(x_{i}c) − (x_{i}c)w = 0,

since x_{i}c is contained in J(A)^{2}, hence commutes with w. Thus Im(f) is annihilated by J(Z(A)), implying Im(f) ⊆ soc(Z(A))∩soc^{2}(A). Since also Im(f) ⊆ [A, A], which intersects soc(A) trivially as A is symmetric split local, statement (ii) follows. □

For monomial algebras, the Lie algebra structure of HH^{1} has been calculated in work of Strametz [19]. Maximal diagonalisable Lie subalgebras of HH^{1} have been calculated by Le Meur [12] for certain algebras without oriented cycles. The dimension of dim(HH^{1}(A)) is related to combinatorial data of the quiver of A in work of de la Peña and Saorín [16].

4. The dimension of HH^{1}(A)

Let k be a field of odd prime characteristic p, let 1 ≠ q ∈ k^{×} have order e dividing p − 1, and let

A = k⟨x, y | x^{p} = y^{p} = 0, yx = qxy⟩.

Then A is a symmetric local k-algebra of dimension p^{2}, having the set of monomials

V = {x^{i}y^{j} | 0 ≤ i, j ≤ p − 1}

as a k-basis. The linear map A → k sending x^{p−1}y^{p−1} to 1 and all other monomials in V to 0 is a symmetrising form for A.

Remark 4.1. One can define an algebra A as above for arbitrary q ∈ k^{×}, but unless the order of q divides p − 1, this yields a selfinjective algebra which is not symmetric. Indeed, if A is symmetric, then any symmetrising form s of A is nonzero on the socle element x^{p−1}y^{p−1}. Thus

0 ≠ s(x^{p−1}y^{p−1}) = s(x^{p−2}y^{p−1}x) = q^{p−1}s(x^{p−1}y^{p−1}),

and hence q^{p−1} = 1. Thus the algebras arising for q not of order dividing p−1 are not Morita equivalent to block algebras of finite groups.

The purpose of this section is to determine the dimension of HH^{1}(A).

Proposition 4.2. We have dim(HH^{1}(A)) = 2(p + (\frac{p−1}{e})^{2}).

We start with some technical observations. The subset

V′ = {x^{i}y^{j} | 0 ≤ i, j ≤ p − 1, (i, j) ≠ (0, 0)}

of A is a k-basis of J(A), and the element x^{p−1}y^{p−1} spans soc(A). For r ≥ 0 the subset

V_{r} = {x^{i}y^{j} | 0 ≤ i, j ≤ p − 1, i + j ≥ r}

of V is a k-basis of J(A)^{r}

Lemma 4.3.

(i) The set {x^{i}y^{j} | 0 ≤ i, j ≤ p − 1, i and j divisible by e, or i = p − 1, or j = p − 1} is a k-basis of Z(A). In particular, we have

dim_{k}(Z(A)) = (\frac{p−1}{e})^{2} + 2p − 1.

(ii) The set {x^{i}y^{p−1}, x^{p−1}y^{j} | p−e ≤ i, j ≤ p−1} is a k-basis of soc(Z(A)); in particular,

we have dim_{k}(soc(Z(A))) = 2e − 1. (iii) We have J(Z(A)) ⊆ J(A)^{e}, and if e = 2, then soc(Z(A)) = soc^{2}(A) ⊆ J(A)^{2}.

Proof. If x and y commute with a linear combination of monomials in the set V, then x and y commute with the monomials with nonzero coefficients in that linear combination. Thus Z(A) has a basis which is a subset of V. Clearly x, y commute exactly with the monomials x^{i}y^{j} where either both i, j are divisible by e or one of i, j is p − 1. This shows that Z(A) has a basis as stated in (i). Since x^{e} and y^{e} are in J(Z(A)), hence annihilate soc(Z(A)), it follows that soc(Z(A)) is contained in the span of the elements of

{x^{i}y^{j} | p − e ≤ i, j ≤ p − 1} ∩ Z(A) = {x^{i}y^{p−1}, x^{p−1}y^{j} | p − e ≤ i, j ≤ p − 1}.

On the other hand, every element of {x^{i}y^{p−1}, x^{p−1}y^{j} | p − e ≤ i, j ≤ p − 1} is annihilated by J(Z(A)), whence (ii). Statement (iii) follows easily from the previous statements. □

Lemma 4.4. The set

{x^{i}y^{j} | 1 ≤ i, j ≤ p − 1, i or j not divisible by e}

is a k-basis of [A, A]. In particular, we have

dim_{k}([A, A]) = (p − 1)^{2} − (\frac{p−1}{e})^{2}

and the space [A, A] is contained in the ideal Axy = xyA.

Proof. Let 1 ≤ i, j ≤ p − 1. If j is not divisible by e, then

[x, x^{i−1}y^{j}] = xx^{i−1}y^{j} − x^{i−1}y^{j}x = (1 − q^{j})x^{i}y^{j} ≠ 0

whence x^{i}y^{j} ∈ [A, A]. Similarly, if i is not divisible by e, then

x^{i}y^{j} = (1 − q^{i})^{−1}[y, x^{i}y^{j−1}] ∈ [A, A].

Thus the given set is contained in [A, A] and it spans a subspace of [A, A] of dimension (p − 1)^{2} − (\frac{p−1}{e})^{2}. Since dim_{k}([A, A]) = dim_{k}(A) − dim_{k}(Z(A)), the formula for dim_{k}([A, A]) follows from Lemma 4.3. This dimension coincides with the dimension of the subspace spanned by the given set, whence the result. □

Let f: A → A be a derivation. Then f(1) = 0, and f is uniquely determined by its values at x and y. An easy induction shows that for any positive integer n and x_{1}, x_{2}, …, x_{n} ∈ A, we have

f(x_{1}x_{2} ⋯ x_{n}) = ∑_{i=1}^{n} x_{1}x_{2} ⋯ x_{i−1}f(x_{i})x_{i+1} ⋯ x_{n} ;

in particular, for any x ∈ A we have f(x^{n}) = ∑_{i=1}^{n} x^{i−1}f(x)x^{n−i}.

Lemma 4.5. For 0 ≤ i, j ≤ p − 1, let α_{i,j}, β_{i,j} ∈ k. There is a derivation f: A → A satisfying

f(x) = ∑_{0≤i,j≤p−1} α_{i,j}x^{i}y^{j}, f(y) = ∑_{0≤i,j≤p−1} β_{i,j}x^{i}y^{j}

if and only if the following hold.

(1) α_{i,j−1}(1 − q^{i−1}) + β_{i−1,j}(1 − q^{j−1}) = 0 for 1 ≤ i, j ≤ p − 1.

(2) α_{0,j−1} = 0 for 1 ≤ j ≤ p − 1.

(3) β_{i−1,0} = 0 for 1 ≤ i ≤ p − 1. In particular, if f is a derivation on A, then f maps J(A) to J(A).

Proof. Suppose that f is a derivation with the given values for x and y. In the following sums, unless otherwise indicated, the indices i and j run from 0 to p − 1. We have

0 = f(0) = f(qxy − yx) = qf(x)y − yf(x) + qxf(y) − f(y)x

= ∑_{i,j} α_{i,j}(qx^{i}y^{j+1} − yx^{i}y^{j}) + ∑_{i,j} β_{i,j}(qx^{i+1}y^{j} − x^{i}y^{j}x)

= ∑_{i,j} α_{i,j}(q − q^{i})x^{i}y^{j+1} + ∑_{i,j} β_{i,j}(q − q^{j})x^{i+1}y^{j}.

The term in the first sum with j = p − 1 is zero, as is the term in the second sum with i = p − 1 (this is where we use the p-power relations x^{p} = 0 = y^{p}). We reindex the first sum with j running from 1 to p − 1, and the second sum with i running from 1 to p − 1, and we then separate the terms with i = 0 or j = 0. This yields

0 = (q − 1) (∑_{j=1}^{p−1} α_{0,j−1}y^{j} + ∑_{i=1}^{p−1} β_{i−1,0}x^{i}) + q ∑_{i,j=1}^{p−1} (α_{i,j−1}(1 − q^{i−1}) + β_{i−1,j}(1 − q^{j−1}))x^{i}y^{j}.

Since q ≠ 1, the first two sums above yield the conditions (2) and (3). The third sum yields the condition (1). In particular, α_{0,0} = β_{0,0} = 0, and hence f(x), f(y) ∈ J(A). This implies that f sends J(A) to J(A).

Conversely, there is a derivation g from the free algebra k⟨x, y⟩ in two generators (abusively again denoted x and y) to the algebra A which takes on x and y the values as given in the statement. By construction, g vanishes on qxy − yx. The properties (1), (2) and (3) imply that g vanishes also on x^{p} and y^{p}. Thus g induces a derivation on A with the required values for x and y. □

Lemma 4.6. We have dim_{k}(Der(A)) = p^{2} + 1 + (\frac{p−1}{e})^{2}.

Proof. A derivation f: A → A is determined by the 2p^{2} coefficients α_{i,j}, β_{i,j} as in Lemma 4.5. Any assignment of values f(x), f(y) satisfying the conditions (1) to (3) in that lemma determines a unique derivation. If e divides both i − 1 and j − 1, then the condition (1) is trivially satisfied, otherwise (1) yields a relation. Thus the condition (1) yields (p−1)^{2}−(\frac{p−1}{e})^{2} relations. The conditions (2) and (3) each yield p − 1 relations. Thus, the total number of relations from Lemma 4.5 is

(p − 1)^{2} − (\frac{p−1}{e})^{2} + 2(p − 1) = p^{2} − 1 − (\frac{p−1}{e})^{2}

and it follows that dim_{k}(Der(A)) = p^{2} + 1 + (\frac{p−1}{e})^{2}. □

Proof of Proposition 4.2. We have dim_{k}(A) = p^{2} and dim_{k}(Z(A)) = (\frac{p−1}{e})^{2} + 2p − 1 from Lemma 4.3. Thus dim_{k}(IDer(A)) = p^{2} − (\frac{p−1}{e})^{2} − 2p + 1. It follows from Lemma 4.6 that

dim_{k}(HH^{1}(A)) = 2 (\frac{p−1}{e})^{2} + 2p,

which completes the proof of Proposition 4.2. □

5. The Lie algebra structure of HH^{1}(A)

A Lie subalgebra H of a Lie algebra L is called toral if the image of H in the adjoint representation on L is simultaneously diagonalisable (hence abelian). For semisimple complex Lie algebras, the maximal toral Lie subalgebras are exactly the Cartan subalgebras. As in the previous section, let k be a field of odd prime characteristic p, let 1 ≠ q ∈ k^{×} have order e dividing p − 1, and let

A = k⟨x, y | x^{p} = y^{p} = 0, yx = qxy⟩.

The technicalities needed for the proof of Theorem 1.1 are contained in the following series of lemmas. We start by identifying innner derivations.

Lemma 5.1. For i, j such that 0 ≤ i, j ≤ p−1, consider the inner derivation d_{i,j} = [x^{i}y^{j}, −] on A.

(i) We have d_{i,j}(x) = (q^{j} − 1)x^{i+1}y^{j}, where 0 ≤ i, j ≤ p − 1. In particular, we have d_{i,j}(x) = 0 if and only if i = p − 1 or e divides j.

(ii) We have d_{i,j}(y) = (1 − q^{i})x^{i}y^{j+1}, where 0 ≤ i, j ≤ p − 1. In particular, we have

d_{i,j}(y) = 0 if and only if j = p − 1 or e divides i. (iii) Let d be an inner derivation of A. Then d(x) is a linear combination of monomials

x^{i}y^{j} with 1 ≤ i, j ≤ p − 1 such that e does not divide j. Similarly, d(y) is a linear combination of monomials x^{i}y^{j} with 1 ≤ i, j ≤ p − 1 such that e does not divide i.

Proof. Let i, j be integers such that 0 ≤ i, j ≤ p − 1. We have

d_{i,j}(x) = [x^{i}y^{j}, x] = x^{i}y^{j}x − x^{i+1}y^{j} = (q^{j} − 1)x^{i+1}y^{j}.

This expression vanishes precisely if q^{j} = 1 or if i + 1 = p, whence (i). As similar calculation proves (ii). An inner derivation on A is a linear combination of the inner derivations d_{i,j}, where 0 ≤ i, j ≤ p − 1. Thus (iii) follows from (i) and (ii). □

Using Lemma 4.5, we determine all derivations on A mapping one of the generators to a single monomial and the other to zero.

Lemma 5.2. Let a, b, c, d be integers such that 0 ≤ a, b, c, d ≤ p − 1.

(i) There is a derivation f_{a,b} on A satisfying f_{a,b}(x) = x^{a}y^{b} and f_{a,b}(y) = 0 if and only if b = p − 1 or a ≥ 1 and e divides a − 1. Moreover, in that case we have

f_{a,b}(x^{c}y^{d}) = (∑_{s=0}^{c−1} q^{bs}) x^{a+c−1}y^{b+d} ,

with the convention that this is zero if c = 0. In particular, if e divides a − 1 and b or if b = p − 1, then

f_{a,b}(x^{c}y^{d}) = cx^{a+c−1}y^{b+d} .

(ii) There is a derivation g_{a,b} on A satisfying g_{a,b}(x) = 0 and g_{a,b}(y) = x^{a}y^{b} if and only if a = p − 1 or b ≥ 1 and e divides b − 1. Moreover, in that case we have

g_{a,b}(x^{c}y^{d}) = (∑_{t=0}^{d−1} q^{at}) x^{a+c}y^{b+d−1} ,

with the convention that this is zero if d = 0. In particular, if e divides a and b − 1, or if a = p − 1, then

g_{a,b}(x^{c}y^{d}) = dx^{a+c}y^{b+d−1} .

Proof. With the notation of Lemma 4.5, the condition f_{a,b}(y) = 0 is equivalent to the vanishing of all coefficients β_{i,j}, where 0 ≤ i, j ≤ p − 1. The condition f_{a,b}(x) = x^{a}y^{b} is equivalent to α_{a,b} = 1 and the vanishing of all remaining coefficients α_{i,j}. If 1 ≤ a ≤ p − 1 and 0 ≤ b ≤ p−2, then the relation (1) from Lemma 4.5 yields 0 = α_{a,b}(1−q^{a−1}) = 1−q^{a−1}, hence that e divides a − 1. If a = 0, then relation (2) from Lemma 4.5 forces b = p − 1. Suppose now that f_{a,b} is a derivation; that is, b = p−1 or a ≥ 1 and e divides a−1. Since f_{a,b}(y) = 0 = f_{a,b}(1), an easy induction shows that f_{a,b}(y^{d}) = 0. Thus f_{a,b}(x^{c}y^{d}) = f_{a,b}(x^{c})y^{d}. Another straighforward induction shows that f_{a,b}(x^{c}) = (∑_{s=0}^{c−1} q^{bs})x^{a+c−1}y^{b}. Combining these facts yields the first formula in (i). If in addition e divides b, then q^{b} = 1, whence the second formula. This proves (i), and the proof of (ii) is similar. □

Note the slight redundancy in the statement of Lemma 5.2: if 0 ≤ a ≤ p − 1 and e divides a − 1, then necessarily a ≥ 1, since we assume that e ≠ 1. We determine next a linearly independent subset of Der(A) whose image in HH^{1}(A) is a k-basis.

Lemma 5.3. Let a, b, c, d be integers such that 0 ≤ a, b, c, d ≤ p − 1. Let X be the disjoint union of the two sets of derivations

{f_{a,b} | 0 ≤ a, b ≤ p − 1, e divides a − 1 and b, or b = p − 1}
{g_{a,b} | 0 ≤ a, b ≤ p − 1, e divides a and b − 1, or a = p − 1}

The set X is linearly independent, and its span H is a complement of IDer(A) in Der(A).

Proof. By Lemma 5.2, the set X indeed consists of derivations. The linear independence of the set X of derivations follows immediately from the fact that the set V of monomials in x and y is a basis of A. The cardinality of the set X is equal to dim_{k}(HH^{1}(A)), by Proposition 4.2. Any nonzero linear combination of the derivations in X map either x or y to a nonzero element in A. If x is mapped to a nonzero element, this element involves a monomial x^{a}y^{b} with b divisible by e. But then Lemma 5.1 implies that this linear combination is not an inner derivation. A similar argument applies if y is mapped to a nonzero element. This shows that the space H spanned by X intersects IDer(A) trivially. Since dim_{k}(H) = |X| = dim_{k}(HH^{1}(A)), it follows that H is a complement of IDer(A) in Der(A). □

We calculate next the Lie brackets between the elements of the basis X of H.

Lemma 5.4. With the notation of Proposition 5.3, let a, b, c, d be integers such that 0 ≤ a, b, c, d ≤ p − 1.

(i) Suppose that e divides a − 1 and b, or that b = p − 1; similarly, suppose that e divides c − 1 and d, or that d = p − 1. If a + c − 1 ≤ p − 1 and b + d ≤ p − 1, then

[f_{a,b}, f_{c,d}] = (c − a)f_{a+c−1,b+d}

and we have b + d = p − 1 or e divides both a + c − 2 and b + d; in particular, we have f_{a+c−1,b+d} ∈ X. If one of a + c − 1 or b + d is at least p, then

[f_{a,b}, f_{c,d}] = 0 .

(ii) Suppose that e divides a and b − 1, or that a = p − 1; similarly, suppose that e divides c and d − 1, or that c = p − 1. If a + c ≤ p − 1 and b + d − 1 ≤ p − 1, then

[g_{a,b}, g_{c,d}] = (d − b)g_{a+c,b+d−1}

and we have a + c = p − 1 or e divides both a + c and b + d − 2; in particular, we have g_{a+c,b+d−1} ∈ X. If one of a + c, b + d − 1 is at least p, then

[g_{a,b}, g_{c,d}] = 0 .

(iii) Suppose that e divides a − 1 and b, or that b = p − 1; similarly, suppose that e divides c and d − 1, or that c = p − 1. If a + c > p − 1 or b + d > p − 1, then

[f_{a,b}, g_{c,d}] = 0 .

(iv) Suppose that e divides a − 1 and b, or that b = p − 1; similarly, suppose that e divides c and d − 1, or that c = p − 1. Suppose that a + c ≤ p − 1 and that b + d ≤ p − 1. We have a + c < p − 1 if and only if b + d < p − 1, and in that case, we have

[f_{a,b}, g_{c,d}] = −bf_{a+c,b+d−1} + cg_{a+c−1,b+d} .

(v) Suppose that e divides a − 1 and b, or that b = p − 1; similarly, suppose that e divides c and d − 1, or that c = p − 1. Suppose that a + c ≤ p − 1 and that b + d ≤ p − 1. We have a + c = p − 1 if and only if b + d = p − 1, and in that case we have (a, b, c, d) = (0, p − 1, p − 1, 0), and

[f_{0,p−1}, g_{p−1,0}] = (q^{−1} − 1)^{−1}[x^{p−2}y^{p−2}, −] = (q^{−1} − 1)^{−1}d_{p−2,p−2} .

Proof. With the assumptions as in (i), both sides vanish at y, and we need to show that they coincide at x. It follows from Lemma 5.2 (i) that

[f_{a,b}, f_{c,d}](x) = f_{a,b}(x^{c}y^{d}) − f_{c,d}(x^{a}y^{b}) = cx^{a+c−1}y^{b+d} − ax^{a+c−1}y^{b+d}

This is a nonzero derivation only if a + c − 1 ≤ p − 1 and b + d ≤ p − 1. If b + d < p − 1, then b < p − 1 and d < p − 1, hence a − 1, c − 1, b, d are divisible by e, and therefore a + c − 2 and b+d are divisible by e. This shows (i), and the proof of (ii) is similar. With the assumptions as in (iii), we have

[f_{a,b}, g_{c,d}](x) = f_{a,b}(g_{c,d}(x)) − g_{c,d}(f_{a,b}(x)) = −g_{c,d}(x^{a}y^{b}) = −bx^{a+c}y^{b+d−1} where the last equation uses Lemma 5.2 (ii). A similar calculation yields [f_{a,b}, g_{c,d}](y) = cx^{a+c−1}y^{b+d}, whence (iii). If a + c = p − 1, then e divides a (since e divides c and p − 1), so e does not divide a − 1, and hence b = p − 1. The hypothesis b + d ≤ p − 1 forces d = 0, so e does not divide d − 1, and hence hence c = p − 1, which in turn forces a = 0 by the hypothesis a + c ≤ p − 1. This shows that under the assumptions in (iv) and (v), we have a + c = p − 1 if and only if b + d = p − 1, which in turn holds if and only if (a, b, c, d) = (0, p − 1, p − 1, 0). For the proof of (iv), assume that a + c < p − 1 and b + d < p − 1. Then all of a, b, c, d are strictly smaller than p − 1. Thus e divides a − 1, d − 1, b, and c, and therefore e divides b + d − 1 and a + c − 1. Hence f_{a+c,b+d−1} and g_{a+c−1,b+d} are in X. We have

[f_{a,b}, g_{c,d}](x) = f_{a,b}(0) − g_{c,d}(x^{a}y^{b}) = −bx^{a+c}y^{b+d−1} ,

where the last equation is from Lemma 5.2 (ii). This is equal to − bf_{a+c,b+d−1} + cg_{a+c−1,b+d} evaluated at x. Similarly,

[f_{a,b}, g_{c,d}](y) = f_{a,b}(x^{c}y^{d}) − g_{c,d}(0) = cx^{a+c−1}y^{b+d} ,

where the last equation is from Lemma 5.2 (i). This is equal to − bf_{a+c,b+d−1} + cg_{a+c−1,b+d} evaluated at y. The formula in (iv) follows. In order to prove (v), we need to calculate

[f_{0,p−1}, g_{p−1,0}](x) = f_{0,p−1}(0) − g_{p−1,0}(y^{p−1}) = x^{p−1}y^{p−2} ,

where the last equation uses Lemma 5.2 (ii) and −(p − 1) = 1 in k. Similarly, we have

[f_{0,p−1}, g_{p−1,0}](y) = f_{0,p−1}(x^{p−1}) − g_{p−1,0}(0) = −x^{p−2}y^{p−1} .

Note that q^{p−2} = q^{−1} since e divides p − 1. By Lemma 5.1, we have

d_{p−2,p−2}(x) = (q^{−1} − 1)x^{p−1}y^{p−2}, d_{p−2,p−2}(y) = −(q^{−1} − 1)x^{p−2}y^{p−1}.

Statement (v) follows. □

The space H in Lemma 5.3 is not a Lie subalgebra of Der(A) because of the relation (v) in 5.4; this relation implies that the images of f_{0,p−1} and g_{p−1,0} in HH^{1}(A) commute (because their Lie bracket is an inner derivation). The Lie brackets between basis elements in X determine the Lie algebra structure of HH^{1}(A). In order to describe this structure, we first identify those elements in X which are commutators.

Lemma 5.5. Let a, b, c, d be integers such that 0 ≤ a, b, c, d ≤ p − 1.

(i) If e divides a − 1 and b, or if b = p − 1, then

[f_{1,0}, f_{a,b}] = (a − 1)f_{a,b} and [g_{0,1}, f_{a,b}] = bf_{a,b} .

In particular, if e divides b, then

[f_{1,0}, f_{1,b}] = 0 and [g_{0,1}, f_{1,b}] = bf_{1,b} .

(ii) Suppose that e divides c and d − 1, or that c = p − 1. We have

[f_{1,0}, g_{c,d}] = cg_{c,d} and [g_{0,1}, g_{c,d}] = (d − 1)g_{c,d} .

In particular, if e divides c, then

[f_{1,0}, g_{c,1}] = cg_{c,1} and [g_{0,1}, g_{c,1}] = 0 .

(iii) The linear endomorphisms ad(f_{1,0}) and ad(g_{0,1}) of Der(A) restrict to linear endomorphisms of the subspace H spanned by X, and, with respect to the basis X, these endomorphisms of H are represented by diagonal matrices.

(iv) All basis elements in X except f_{1,0} and g_{0,1} are commutators in Der(A).

(v) We have [f_{1,0}, g_{0,1}] = 0.

Proof. The statements (i) and (ii) are special cases of Lemma 5.4, and the statements (iii), (iv), and (v) follow from (i) and (ii). □

Lemma 5.6. Let a, b, c, d be integers such that 0 ≤ a, b, c, d ≤ p − 1.

(i) Suppose that e divides a − 1 and b, or that b = p − 1. If a + b ≥ 2, then a ≥ e + 1 or b ≥ e. In particular, a + b − 1 ≥ min{e, p − 2}.

(ii) Suppose that e divides c and d − 1, or that c = p − 1. If c + d ≥ 2, then c ≥ e or d ≥ e + 1. In particular, c + d − 1 ≥ min{e, p − 2}.

Proof. Assume that b < e. Then b < p − 1, so e divides a − 1 and b. The inequality b < e forces b = 0. Since a + b ≥ 2, this implies a ≥ 2, hence a − 1 ≥ 1. Since e divides a − 1, it follows that a − 1 ≥ e, and hence a + b − 1 ≥ e. If b ≥ p − 1, then a + b − 1 ≥ p − 2, whence (i). A similar argument yields (ii). □

Proof of Theorem 1.1. Statement (i) is proved in Proposition 4.2. We use the same notation as in Theorem 1.1; in particular, L = HH^{1}(A) and L′ is the derived Lie subalgebra of L. It follows from Lemma 5.5 that L′ contains the images of all elements of X except possibly the images of f_{1,0} and g_{0,1}.

The relations in Lemma 5.4 imply that L′ contains no nonzero linear combination of the images of f_{1,0} and g_{0,1}. Thus L′ has codimension 2 in L.

A complement of L′ is spanned by the image of {f_{1,0}, g_{0,1}}, and this complement is a 2-dimensional abelian Lie subalgebra of L, by Lemma 5.5 (v). Moreover, L′ has as a basis the image in HH^{1}(A) of the set

X′ = X \ {f_{1,0}, g_{0,1}} .

Equivalently, X′ consists of all f_{a,b}, g_{c,d} in X with a + b ≥ 2 and c + d ≥ 2.

It follows from Lemma 5.5 (iii) that the images of f_{1,0} and g_{0,1} span a toral subalgebra. Lemma 5.5 implies that the centraliser in L of the image of f_{1,0} is spanned by the images of f_{1,b}, g_{0,d} with b and d − 1 divisible by e. Similarly, the centraliser in L of the image of g_{0,1} is spanned by the images of f_{a,0}, g_{c,1}, with a−1 and c divisible by e. Thus Z(L) is contained in the span of the images of f_{1,0} and g_{0,1}, but it follows again from Lemma 5.5 that no nonzero linear combination of these two elements is in the center. This shows that Z(L) = {0} and that the toral subalgebra H is maximal. This proves (ii) and (iii).

For m ≥ 1 denote by L_{m} the subspace of L spanned by the images of those f_{a,b}, g_{c,d} in X for which a + b ≥ m and c + d ≥ m. Thus L_{1} = L, L_{2} = L′, and L_{m} = {0} for m ≥ 2p. The relations in Lemma 5.4 imply that [L′, L_{m}] ⊆ L_{m+1}, which in turn implies that L′ is nilpotent, whence (iv).

The socle of L as a Z(A)-module is contained in the subspace of L which is annihilated by x^{e} and y^{e}. We have x^{e}f_{a,b} = f_{a+e,b} if a + e ≤ p − 1, and x^{e}f_{a,b} = 0 if a ≥ p − e. Similarly, we have y^{e}f_{a,b} = f_{a,b+e} if b + e ≤ p − 1 and y^{e}f_{a,b} = 0 if b ≥ p − e. It follows that the socle of L as a Z(A)-module is equal to the subspace of HH^{1}(A) which is annihilated by x^{e} and y^{e}. Thus the image of f_{a,b} in L is contained in soc_{Z(A)}(L) if a ≥ p − e and b ≥ p − e. Since also e divides both a − 1 and b or b = p − 1, this forces b = p − 1. Similarly, the image of g_{a,b} in L is contained in soc_{Z(A)}(L) if and only if b ≥ p and a = p − 1. It follows that soc_{Z(A)}(L) is equal to the space spanned by the image in L of the set

S = {f_{a,p−1} | p − e ≤ a ≤ p − 1} ∪ {g_{p−1,b} | p − e ≤ b ≤ p − 1}

This shows in particular that

dim_{k}(soc_{Z(A)}(L)) = 2e .

The relations in Lemma 5.4 imply that we have [X′, S] = {0}, and hence we have an inclusion

soc_{Z(A)}(L) ⊆ Z(L′) .

This proves (v).

By the above and Lemma 5.6, L′ is spanned by the images of elements f_{a,b}, g_{c,d} where at least one of a, b is greater or equal to e, and where at least one of c, d is greater or equal to e. Thus L′ is contained in x^{e}L + y^{e}L. Since no nonzero linear combination of the images of f_{1,0}, g_{0,1} is contained in J(Z(A))L, statement (vi) follows.

In order to prove (vii), suppose first that e = p − 1. In that case we have

X′ = {f_{a,p−1} | 0 ≤ a ≤ p − 1} ∪ {g_{p−1,b} | 0 ≤ b ≤ p − 1}

The images in L of any two elements of X′ commute; more precisely, any two elements in X′ commute already in H, except for [f_{0,p−1}, g_{p−1,0}], which is inner by Lemma 5.4 (v). This shows that L′ is abelian if e = p − 1.

Suppose that e < p − 1; in particular, e ≤ \frac{p−1}{2}. We consider the basis X′ = X \ \{f_{1,0}, g_{0,1}\} of L′. Since e < p − 1, there are derivations f_{e+1,0}, g_{0,e+1}, f_{1,e}, and g_{e,1} in X′. Using Lemma 5.4, one verifies that the Lie brackets of any of these four elements with any element in X′ yield elements in X, possibly multiplied by scalars (which can be zero). It follows that in order to calculate centralisers in L′ of these four particular elements, it suffices to calculate centralisers in the space H′ spanned by X′. It follows further that if one of the above four elements centralises a linear combination of elements in X′, it centralises the elements of X′ with nonzero coefficients individually. A tedious verification, using Lemma 5.4, shows that the centraliser of f_{e+1,0} in X′ intersected with the centraliser of g_{0,e+1} is the set S_{1} ∪ S_{2}, where

S_{1} = {f_{a,p−1} | p − e ≤ a ≤ p − 1} ∪ {f_{e+1,0}, f_{p−e,0}, f_{e+1,p−1}}

and

S_{2} = {g_{p−1,d} | p − e ≤ d ≤ p − 1} ∪ {g_{0,e+1}, g_{0,p−e}, g_{p−1,e+1}}

The element f_{1,e} does not centralise any of the two elements f_{e+1,0} and f_{p−e,0}. Similarly, g_{e,1} centralises neither g_{0,e+1} nor g_{0,p−e}. Thus every element in Z(L′) is the image of a linear combination of the set

S_{3} = {f_{a,p−1} | p − e ≤ a ≤ p − 1} ∪ {f_{e+1,p−1}} ∪ {g_{p−1,d} | p − e ≤ d ≤ p − 1} ∪ {g_{p−1,e+1}}

One verifies that the image of S_{3} is contained in Z(L′). The cardinality of S_{3} is 2e + 2. Statement (vii) follows.

For the last statement, note that the p-power map on L is induced by the map sending a derivation f on A to the composition f^{[p]} = f ∘ f ∘ ⋯ ∘ f of f with itself p times. We clearly have (f_{1,0})^{[p]} = f_{1,0}, and (g_{0,1})^{[p]} = g_{0,1}; that is, the images of f_{1,0} and g_{0,1} in L are p-toral. Since the image of {f_{1,0}, g_{0,1}} in L is a basis of H, this shows that H is p-toral. Any element of X′ = X \ {f_{1,0}, g_{0,1}} is of the form f_{a,b} or g_{a,b} with a + b − 1 ≥ e or a + b − 1 ≥ p − 2 (the latter arises if a or b is equal to p − 1). Consider first the case where p ≥ 5, so that p − 2 ≥ 3. Since e ≥ 2, it follows that a + b − 1 ≥ 2. Lemma 5.2 implies that any derivation in X′ sends a monomial in x, y of total degree m to a scalar multiple of a monomial of total degree at least m + 2. Thus any composition of p elements in X′ sends a monomial in x, y of degree at least 1 to a scalar multiple of a monomial x^{c}y^{d} of total degree c + d ≥ 1 + 2p. This implies that at least one of c, d is greater that p, which in turn implies that x^{c}y^{d} = 0 in A. It follows that any composition of p elements in X′ is zero. Therefore, if f is a linear combination of elements in X′, then f^{[p]} = 0. Since the image in L of X′ is a basis of L′, this proves (viii) in the case p ≥ 5. If p = 3, then e = 2, and we have

X′ = {f_{0,2}, f_{1,2}, f_{2,2}, g_{2,0}, g_{2,1}, g_{2,2}} .

A direct verification shows that the composition of any three derivations in this set is zero, completing the proof. □

Proof of Corollary 1.3. A stable equivalence of Morita type preserves the Tate analogue of Hochschild cohomology, hence preserves HH^{1}(A) as a module over HH^{0}(A) ≅ Z(A) since the projective ideal in Z(A) annihilates Hochschild cohomology in positive degrees. The corollary follows from statement (v) in Theorem 1.1 together with Corollary 3.2. □

Proof of Corollary 1.4. First consider the case that B is nilpotent. By the structure theorem of nilpotent blocks [17, (1.4.1)], B is a matrix algebra over kP. Since P is either cyclic or elementary abelian of order p^{2}, dim_{k}(J(B)/J(B)^{2}) equals 1 or 2, and the result holds. Thus, we may assume that B is not nilpotent. In particular, since P is abelian, I > 1 [4, (1.ex.3)]. If P is cyclic, then since C has a unique isomorphism class of simple modules, I = 1, a contradiction. Thus, we may assume that P is elementary abelian. By [9, Theorem 1.1], the inertial quotient of B is abelian. By the structure theory of blocks with normal defect group ([11, Theorem A] or [20, §45]), C is a matrix algebra over a twisted group algebra of the semidirect product of P with the inertial quotient of B. Hence, since C has a unique isomorphism class of simple modules, by [6, Lemma 2], the inertial quotient of B is a direct product of two cyclic groups of order \sqrt{I} (see for instance the proof of Theorem 1.1 and Proposition 5.3 of [9]). By Theorem 4.2 and Corollary 4.3 of [7] and their proofs, a basic algebra of C is isomorphic to the algebra A of Theorem 1.1 with 1 < e ≤ \sqrt{I}. By [9, Theorem 1.1], B is local. Finally, by [13, Theorem A.2], there is a stable equivalence of Morita type between B and C. The result now follows from Corollary 1.3. □

6. Lifting quantum complete intersections over O

Let p be an odd prime and O a complete discrete valuation ring containing a primitive 4p-th root of unity, with residue field k of characteristic p and field of fractions K of characteristic 0.

We denote in this section by G a finite group obtained as a semi-direct product of an elementary abelian p-group

P = ⟨g⟩ × ⟨h⟩ ≅ C_{p} × C_{p}

of rank two by a quaternion group Q_{8} = ⟨s, t | s^{4} = 1, s^{2} = t^{2}, sts^{3} = t^{3}⟩ of order 8, acting on P by sgs^{−1} = g^{−1}, shs^{−1} = h, tgt^{−1} = g, and tht^{−1} = h^{−1}. In particular, the unique central involution z = s^{2} = t^{2} of Q_{8} acts trivially on P, hence Z(G) = ⟨z⟩. The group algebra OG has two blocks, the principal block B_{0} = OGe_{0}, where e_{0} = \frac{1}{2}(1 + z), and one nonprincipal block B_{1} = OGe_{1}, where e_{1} = \frac{1}{2}(1 − z). The block B_{1} has a unique isomorphism class of simple modules, and more precisely, the quantum complete intersection

A = k⟨x, y | x^{p} = y^{p} = 0, xy + yx = 0⟩

is a basic algebra of k⊗_{O} B_{1}. We determine the structure of a basic algebra of B_{1}. To do this, we will require the Chebyshev polynomials T_{n} of the first kind. For n ≥ 0, the polynomial T_{n} in the variable u is the unique polynomial in ℤ[u] of degree n satisfying T_{n}(cos(θ)) = cos(nθ) for any θ ∈ ℝ. Using sin(θ) = cos(θ − \frac{π}{2}) we obtain for n odd the formula

sin(nθ) = (−1)^{\frac{n−1}{2}} T_{n}(sin(θ)) .

The polynomials T_{n} can be defined recursively by T_{0}(u) = 1, T_{1}(u) = u, and T_{n+1}(u) = uT_{n}(u) − T_{n−1}(u) for n ≥ 1. This recursion formula shows that the leading coefficient of T_{n} is 2^{n−1}. It also shows that for n even (resp. odd), the polynomial T_{n} is involves only even (resp. odd) powers of the variable u. For n ≥ 0, define a polynomial f_{n} in the variable u by

f_{n}(u) = 2T_{n}(\frac{u}{2}) .

Then f_{0}(u) = 2, f_{1}(u) = u, and f_{n+1}(u) = uf_{n}(u)−f_{n−1}(u). In particular, f_{n} is a polynomial in ℤ[u] with leading coefficient 1, and if n is even (resp. odd), then f_{n} involves only even (resp. odd) powers of u. The well-known explicit formulae for Chebyshev polynomials imply that if n = p, then all coefficients of f_{p} other than the leading coefficient of f_{p} are divisible by p, and hence f_{p} reduces to the monomial u^{p} in k[u].

Theorem 6.1. With the notation above, let Â be the O-algebra

Â = O⟨γ, δ | γδ + δγ = 0, f_{p}(γ) = 0 = f_{p}(δ)⟩

Then Â is a basic algebra of B_{1}; in particular, we have k ⊗_{O} Â ≅ A.

Proof. Since e_{1} annihilates the OQ_{8}-modules of rank one, it follows that S = OQ_{8}e_{1} is the quotient algebra of OQ_{8} corresponding to the unique irreducible character of Q_{8} of degree 2, hence isomorphic to the matrix algebra M_{2}(O). The unique simple B_{1}-module (up to isomorphism) has dimension 2. Thus, setting Â = C_{B_{1}}(S) = B_{1}^{Q_{8}}, we get from [20, (7.5) Proposition] that

B_{1} = S ⊗_{O} Â

and then necessarily Â is a basic algebra of B_{1}, as its unique simple module has dimension 1. We need to show that Â has generators satisfying the relations as in the statement, and then we need to show that there are no other relations. We use the generators g, h, s, t of the group G with the relations as stated at the beginning of this section. Define elements γ and δ in B_{1} by

γ = (g − g^{−1})te_{1} , δ = (h − h^{−1})se_{1} .

Note that t commutes with g, g^{−1}, hence with g − g^{−1}. Similarly, s commutes with h, h^{−1}, and with h − h^{−1}. We have

ste_{1} = \frac{1}{2}st(1 − t^{2}) = \frac{1}{2}s(t − t^{−1})
tse_{1} = \frac{1}{2}st^{3}(1 − t^{2}) = \frac{1}{2}s(t^{−1} − t)

and hence tse_{1} = −ste_{1}. Using this equality, we verify that se_{1} and te_{1} commute with γ and δ. We have

s(g − g^{−1})te_{1} = (g^{−1} − g)ste_{1} = (g − g^{−1})tse_{1}

which shows that se_{1} and γ commute. Similar calculations show the remaining commutation relations. This shows that the elements γ and δ are in Â, and we need to show that they generate Â. Note that g − g^{−1} = (g^{2} − 1)g^{−1} is a generator of J(k⟨g⟩)/J(k⟨g⟩)^{2}; similarly for h. Thus (g − g^{−1})e_{1} and (h − h^{−1})e_{1} generate the radical modulo the radical square of the image of OP in B_{1}, hence these two elements together with e_{1} generate the algebra OP e_{1}. The two elements γ and δ are obtained by multiplying (g −g^{−1})e_{1} and (h−h^{−1})e_{1} by te_{1} and se_{1}, respectively, and the two elements se_{1} and te_{1} generate S as an O-algebra. It follows that the set {se_{1}, te_{1}, γ, δ} generates B_{1} as an O-algebra. But then γ and δ necessarily generate Â as a unitary algebra.

We verify that γ and δ satisfy the relations as stated. We have

γδ = (g − g^{−1})t(h − h^{−1})se_{1} = −(g − g^{−1})(h − h^{−1})tse_{1}
= (h − h^{−1})(g − g^{−1})ste_{1} = −(h − h^{−1})s(g − g^{−1})te_{1} = −δγ

whence the anti-commutation relation for γ and δ. For the remaining relations, we first consider the element g − g^{−1} in O⟨g⟩. This element acts on any O⟨g⟩-module of rank one as multiplication by ζ − ζ^{−1} for some p-th root of unity ζ. This is an imaginary number; writing ζ = e^{\frac{2πm}{p}} for some integer m, we get that ζ − ζ^{−1} = 2 sin(\frac{2πm}{p})τ, where τ satisfies τ^{2} = −1. Thus \frac{τ}{2}(g − g^{−1}) acts as multiplication by − sin(\frac{2πm}{p}). Since T_{p} involves only odd powers of x, it follows that T_{p}(\frac{τ}{2}(g − g^{−1})) acts as multiplication by ± sin(p\frac{2πm}{p}) = 0, and hence T_{p}(\frac{τ}{2}(g − g^{−1})) = 0 in O⟨g⟩, or equivalently, f_{p}(τ(g − g^{−1})) = 0. We calculate the odd powers of γ and δ. For n = 2m + 1 for some integer m ≥ 0 we have

(te_{1})^{n} = t(t^{2}e_{1})^{m} = (−1)^{m}te_{1} = τ^{n−1}te_{1} ,

and hence we have

γ^{n} = (g − g^{−1})^{n}t^{n}e_{1} = τ^{n−1}(g − g^{−1})^{n}te_{1} = τ^{−1}(τ(g − g^{−1}))^{n}te_{1}

Thus, using again that f_{p} involves only odd powers of x, we have

f_{p}(γ) = τ^{−1}f_{p}(τ(g − g^{−1}))te_{1} = 0.

A similar calculation yields f_{p}(δ) = 0. This shows that γ and δ satisfy the relations as stated. That is, Â is a quotient of the unitary O-algebra

C = O⟨γ, δ | γδ + δγ = 0, f_{p}(γ) = 0 = f_{p}(δ)⟩.

As an O-module, Â is free of rank p^{2}. The relations defining C imply that C is generated, as an O-module, by the images of the p^{2} monomials γ^{i}δ^{j}, with 0 ≤ i, j ≤ p − 1, and hence C is, as an O-module, a quotient of a free O-module of rank p^{2}. This forces C ≅ Â, whence the result. □

If B is a nilpotent block of some finite group algebra, then the largest O-free commutative algebra quotient of a basic algebra of B is symmetric. Indeed, in that case the basic algebras of B are isomorphic to OQ for some defect group Q of B, and the largest O-free commutative algebra quotient of OQ is the symmetric O-algebra OQ/Q′, where Q′ is the derived subgroup of Q. In [10, Remark 1.3], the question was raised whether this property characterises nilpotent blocks. Some evidence for this comes from a theorem of Okuyama and Tsushima in [14] which states that B has a commutative (and necessarily symmetric) basic algebra if and only if B is nilpotent with abelian defect groups. For the sake of testing this question, we calculate the largest commutative O-algebra quotient of the basic algebra Â of the nonprincipal (and non-nilpotent) block B_{1} of OG, and show that this is indeed not symmetric.

The irreducible characters of B_{1} have degree either 2 or 4. Thus the simple K ⊗_{O} Âmodules have dimension either 1 or 2. The number of simple K ⊗_{O} Â-modules of dimension 1 is equal to 2p − 1, and this is also equal to the O-rank of the largest O-free commutative quotient of Â. This quotient is of the form Â/I, where I is the smallest O-pure ideal in Â which contains [Â, Â]. Its structure is as follows.

Proposition 6.2. Let Â = O⟨γ, δ | γδ + δγ = 0, f_{p}(γ) = 0 = f_{p}(δ)⟩ as in the previous theorem.

(i) The set {γ^{i}δ^{j} | 1 ≤ i, j ≤ p − 1, i or j odd} is an O-basis of [Â, Â].

(ii) The smallest O-pure ideal in Â which contains [Â, Â] is equal to Âγδ = γδÂ, and the

set {γ^{i}δ^{j} | 1 ≤ i, j ≤ p − 1} is an O-basis of this ideal. (iii) The largest O-free commutative quotient of Â is isomorphic to

D = O⟨μ, ν | μν = νμ = 0, f_{p}(μ) = f_{p}(ν) = 0⟩ .

The set {1} ∪ {μ^{i}, ν^{i} | 1 ≤ i ≤ p − 1} is an O-basis of D; in particular, the O-rank of this quotient is 2p − 1.

(iv) We have k ⊗_{O} D ≅ k⟨μ, ν | μν = νμ = 0, μ^{p} = ν^{p} = 0⟩, and the k-algebra k ⊗_{O} D is not symmetric; in particular, the O-algebra D is not symmetric.

Proof. The algebra K ⊗_{O} Â is split semisimple, and hence rk_{O}(Z(A))+rk_{O}([Â, Â]) = rk_{O}(Â). Since A is symmetric, we have dim_{k}(Z(A)) + dim_{k}([A, A]) = dim_{k}(A). Since the canonical map Z(Â) → Z(A) is surjective, it follows that rk_{O}([Â, Â]) = dim_{k}([A, A]). Thus [Â, Â] is an O-pure O-submodule of Â. The same arguments is in the proof of Lemma 4.4 show that the set {γ^{i}δ^{j} | 1 ≤ i, j ≤ p − 1, i or j odd} is contained in [Â, Â]. This set spans an O-pure O-submodule of Â mapping onto [A, A], and hence this set is an O-basis of [Â, Â]. This proves (i). The ideal generated by the set {γ^{i}δ^{j} | 1 ≤ i, j ≤ p − 1, i or j odd} contains the set {γ^{i}δ^{j} | 1 ≤ i, j ≤ p − 1}. The O-span of the latter is an ideal, whence (ii). It follows from (ii) that A/Aγδ is the largest O-free commutative quotient of Â. The relations of this quotient are obtained from those of Â, whence (iii). The image of the polynomial f_{p}(u) in k[u] is x^{p}. Thus the relations of k ⊗_{O} D follow from those of D. The socle of the k-algebra k ⊗_{O} D contains the images of μ^{p−1} and ν^{p−1}, hence has dimension at least 2. Since k ⊗_{O} D is local, this shows that k ⊗_{O} D is not symmetric, and hence neither is D. □

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