On simple modules over twisted finite category algebras

Markus Linckelmann and Michal Stolorz

April 26, 2011

Abstract

The purpose of this note is to show that the recent proof, by Ganyushkin, Mazorchuk and Steinberg, of the parametrisation of simple modules over finite semigroup algebras due to Clifford, Munn and Ponizovskiĭ carries over to twisted finite category algebras. We observe that the parametrisations of simple modules over Brauer algebras, Temperley-Lieb algebras, and Jones algebras due to Graham and Lehrer, can be obtained as special cases of our main result. We further note that the notion of weights in the context of Alperin’s weight conjecture extends to twisted finite category algebras.

1 Introduction

Let k be a commutative ring and C a small category. The set of idempotents in the morphism set Mor(C) of C is partially ordered via e ≤ f if e, f are idempotent endomorphisms of the same object in C such that e = e ∘ f = f ∘ e. For any idempotent endomorphism e of an object X in C we denote by G_{e} the group of invertible elements in the monoid e ∘ End_{C}(X) ∘ e. Two idempotent endomorphisms e ∈ End_{C}(X) and f ∈ End_{C}(Y) are called isomorphic if there are morphisms s ∈ f ∘ Hom_{C}(X, Y) ∘ e and t ∈ e ∘ Hom_{C}(Y, X) ∘ f such that t ∘ s = e and s ∘ t = f. The elements s and t determine a group isomorphism G_{e} ≅ G_{f}, sending u ∈ G_{e} to s ∘ u ∘ t, with inverse sending v ∈ G_{f} to t∘v ∘s. A different choice of s, t will lead to a possibly different group isomorphism, but any two different choices will yield group isomorphisms which differ at most by an inner automorphism of G_{e}. If U is a kC-module then eU is a module for the monoid algebra k(e ∘ End_{C}(X) ∘ e), hence restricts to a module over the group algebra kG_{e}. Two pairs (e, U), (f, V) consisting of idempotent endomorphisms e ∈ End_{C}(X), f ∈ End_{C}(Y), a kG_{e}-module U and a kG_{f}-module V are called isomorphic if there are morphisms s ∈ f ∘ Hom_{C}(X, Y) ∘ e and t ∈ e ∘ Hom_{C}(Y, X) ∘ f such that t ∘ s = e and s ∘ t = f, and such that the isomorphism classes of U, V correspond to each other through the algebra isomorphism kG_{e} ≅ kG_{f} induced by s and t. Since inner automorphisms of an algebra stabilise all isomorphism classes of modules, this property is independent of the choice of s and t.

Theorem 1.1. Let k be a commutative ring and C a finite category. The map sending a simple kC-module S to the pair (e, eS), where e is a minimal idempotent endomorphism in C satisfying eS ≠ {0}, induces a bijection between the set of isomorphism classes of simple kC-modules and the set of isomorphism classes of pairs (e, T) consisting of an idempotent endomorphism e in C and a simple kG_{e}-module T.

For semigroup algebras this is due to Clifford [3], [4], Munn [16] and Ponizovskiĭ [17]. If C is a finite EI-category (that is, if all endomorphisms of objects are isomorphisms), the above parametrisation specialises to the parametrisation, due to Lück, in terms of simple modules of automorphism groups of objects (see [5, §11, B]), [15, §9], or also [23, 4.3]). For inverse semigroup algebras and inverse category algebras this follows also from [21] and [14]. Our proof of Theorem 1.1 follows closely the lines of that given in [6] for semigroups, which has motivated the present work. As in [6], this is based on Green’s work on Schur functors in [10, 6.2]. More precisely, if e is an idempotent in a k-algebra A, then for any simple A-module S, either eS is zero or eS is a simple eAe-module. This induces a bijection between the set of isomorphism classes of simple A-modules not annihilated by e and the set of isomorphism classes of simple eAe-modules. If T is a simple eAe-module then Ae ⊗_{eAe} T has a unique simple quotient and Hom_{eAe}(eA, T) has a unique simple submodule, both isomorphic to a simple A-module S satisfying eS ≅ T. This yields two explicit descriptions of the inverse map from the set of isomorphism classes of pairs (e, T) as in 1.1 to the set of isomorphism classes of simple kC-modules.

The methods work more generally for twisted finite category algebras. In order to describe this, let C be a finite category, k a commutative ring and α a 2-cocycle of C with coefficients in k^{×}. That is, if s, t are morphisms in C for which the composition t ∘ s is defined, then α(t, s) ∈ k^{×} such that if s , t , u are three morphisms in C for which the compositions t ∘ s and u ∘ t are defined, we have the 2-cocycle identity α(u ∘ t, s)α(u, t) = α(u, t ∘ s)α(t, s). As a k-module, the twisted category algebra k_{α}C is equal to kC, endowed with the k-bilinear multiplication t · s = α(t, s)t ∘ s if t ∘ s is defined, and t · s = 0 if not. The associativity of the multiplication in k_{α}C is equivalent to the 2-cocycle identity. The isomorphism class of k_{α}C depends only on the class of α in H^{2}(C; k^{×}). If e is an idempotent endomorphism of an object X in C, the image of e in k_{α}C need no longer be an idempotent, because e · e = α(e, e)(e ∘ e) = α(e, e)e. Dividing this equation by α(e, e)^{2} shows that ê = α(e, e)^{−1}e is an idempotent in k_{α}C. We use the same letter α for the restriction of α to the monoid e ∘ End_{C}(X) ∘ e and its subgroup G_{e}. We have ê · k_{α}C · ê = k_{α}(e ∘ End_{C}(X) ∘ e); in particular, the twisted group algebra k_{α}G_{e} is a unitary subalgebra of ê · k_{α}C · ê. Therefore, if U is a k_{α}C-module, then êU is a k_{α}G_{e}-module. We will see in 5.2 and 5.4 below that if e, f are isomorphic idempotents in C there is an algebra isomorphism k_{α}G_{e} ≅ k_{α}G_{f} which is uniquely determined up to an innner automorphism. Hence, as before, two pairs (e, U), (f, V) consisting of idempotents e ∈ End_{C}(X), f ∈ End_{C}(Y), a k_{α}G_{e}-module U and a k_{α}G_{f}-module V, are called isomorphic if e, f are isomorphic and if the isomorphism classes of U, V correspond to each other through the induced isomorphism k_{α}G_{e} ≅ k_{α}G_{f}. With this notation, Theorem 1.1 can be generalised as follows.

Theorem 1.2. Let k be a commutative ring and C a finite category. Let α be a 2-cocycle of C with coefficients in k^{×}. The map sending a simple k_{α}C-module S to the pair (e, êS), where e is a minimal idempotent endomorphism in C satisfying êS ≠ {0} and where ê = α(e, e)^{−1}e, induces a bijection between the set of isomorphism classes of simple k_{α}C-modules and the set of isomorphism classes of pairs (e, T) consisting of an idempotent endomorphism e in C and a simple k_{α}G_{e}-module T.

Theorem 1.1 is, of course, simply the special case of Theorem 1.2 applied to α = 1, but in order to separate the rather tedious technicalities related to the 2-cocycle α from the structural arguments underlying the proof, we first prove 1.1 and then adapt this proof to the twisted case.

Remark 1.3. Certain combinatorially defined algebras, such as Brauer algebras, Temperley-Lieb algebras, their cyclotomic analogues, Jones algebras and partition algebras can be interpreted as twisted semigroup algebras; see Wilcox [24]. In conjunction with Theorem 1.2 this yields alternative proofs for the parametrisations of simple modules for these algebras, without referring to their cellularity. Denote by B_{n}(δ) the Brauer algebra over a field k with parameter δ ∈ k^{×}, where n ≥ 2. By a result of Graham and Lehrer [7, (4.17)], the isomorphism classes of simple B_{n}(δ)-modules are parametrised by pairs (t, λ), where t ≥ 0 is an integer such that n−t is nonnegative and even, and where λ is a partition of t (a p-regular partition if the characteristic p of k is positive). This is related to 1.2 as follows. Any integer t as above yields an idempotent e_{t} in the semigroup BR_{n} of diagrams which form a basis of B_{n}(δ); the idempotent e_{t} corresponds to a diagram with t strictly vertical strands and with horizontal strands connecting the remaining nodes (see [24, §8] for more details). This yields a set of representatives of the isomorphism classes of idempotents in the semigroup BR_{n}. Moreover, the group G_{e_{t}} corresponding to the idempotent e_{t} is isomorphic to the symmetric group S_{t}. The 2-cocycle arising in the Brauer algebra yields the trivial class upon restriction to S_{t}. The second component λ corresponds to an isomorphism class of simple kS_{t}-modules. Thus the parametrisation from 1.2 yields that in [7, (4.17)]. The Temperley-Lieb algebra T L_{n}(δ) is the subalgebra of B_{n}(δ) spanned by the subsemigroup T L_{n} of BR_{n} of diagrams in BR_{n} which do not have any intersecting strands. The semigroup T L_{n} contains the idempotents e_{t}, and the coresponding groups G_{e_{t}} are trivial. Thus, by 1.2, the isomorphism classes of simple T L_{n}(δ)-modules are parametrised by the set of integers t ≥ 0 such that n − t ≥ 0 is even; this is [7, (6.8)]. Similarly, one can use 1.2 to obtain the parametrisation of the isomorphism classes of simple modules over the Jones algebra in [7, (6.17)] and over partition algebras in [25, 4.11]. The same approach should also work for the cyclotomic Brauer algebras [11], [20] and cyclotomic Temperley-Lieb algebras [19] with additional parameter m > 0. These algebras have idempotents as above, with associated groups of invertible elements isomorphic to ℤ/mZ ≀S_{t} and direct products of copies of ℤ/mℤ, respectively. Therefore 1.2 should yield again parametrisations, as in [19, 5.4], [20, 5.9], and [11, Prop. 43], of the isomorphism classes of simple modules for these algebras.

Remark 1.4. Twisted fusion category algebras arise in the context of reformulations of Alperin’s weight conjecture in [12], [13]. Theorem 1.2 suggests how to extend the notion of weights to arbitrary twisted finite category algebras: with the notation of 1.2, a weight of k_{α}C is a pair (e, T) consisting of an idempotent in C and a simple and projective k_{α}G_{e}-module T. Denote by w(k_{α}C) the number of isomorphism classes of weights of k_{α}C. The isomorphism classes of weights of k_{α}C correspond to a subset of the isomorphism classes of simple k_{α}C-modules, through the correspondence in 1.2. Thus, if we choose an idempotent c in k_{α}C which annihilates exactly all simple modules which do not correspond to a weight, the resulting algebra c · k_{α}C · c has exactly w(k_{α}C) isomorphism classes of simple modules. This algebra has been considered in the context of finite EI-categories and fusion systems in [12], where it is shown to be quasi-hereditary (cf. [12, 2.4, 4.5]). Combining 1.2 with Alperin’s weight conjecture yields a conjecture on the number of isomorphism classes of simple modules of twisted finite category algebras over algebraically closed fields of positive characteristic.

Remark 1.5. Let C be a finite category and k a complete local Noetherian commutative ring with a residue field of prime characteristic p. Let S be a simple kC-module and e a minimal idempotent in Mor(C) such that eS ≠ {0}. Let (Q, V) be a vertex-source pair for the simple kG_{e}-module eS; that is, Q is a minimal subgroup of G_{e} such that eS is relatively kQ-projective and V is an indecomposable kQ-module such that eS is isomorphic to a direct summand of Ind_{Q}^{G_{e}}(V). By Green’s work in [9], the group Q is a p-group and the pair (Q, V) is uniquely determined up to conjugation in G_{e}; see for instance [2, 3.10] or [22, Chapter 3] for expositions of these concepts. It is tempting to call (Q, V) a vertex-source pair of the simple kC-module S. In the context of finite group algebras, a vertex-source pair can be associated with any indecomposable module. This raises the question how to generalise vertices and sources to arbitrary indecomposable modules over the category algebra kC. For finite EI-categories, Fei Xu has developed in [26] a notion of vertices and sources, where the vertices are certain full subcategories.

2 Green relations for categories

The results in this section are translations of basic results from semigroups to categories. For the most part the proofs, included here for the convenience of the reader, are straightforward adaptations of the proofs in the semigroup case; see, for instance, the exposition in [18, Appendix A]. Let C be a small category, with morphism set denoted Mor(C). For subsets M, M′ of Mor(C) we denote by M ∘ M′ the (possibly empty) subset of Mor(C) consisting of all morphisms of the form s ∘ s′, where s ∈ M and s′ ∈ M′ such that this composition is defined. If M consists of a single morphism s we write also s ∘ M′ instead of M ∘ M′; similarly for M′. We use the obvious generalisation of this notation for the composition of more than two subsets of Mor(C). Following standard notation of the Green relations for semigroups [8], for any s, t ∈ Mor(C) we write s ℒ t if Mor(C) ∘ s = Mor(C) ∘ t, we write s ℛ t if s ∘ Mor(C) = t ∘ Mor(C), and we write s𝒥 t if Mor(C) ∘ s ∘ Mor(C) = Mor(C) ∘ t ∘ Mor(C). A morphism s ∈ Hom_{C}(X, Y) is called split if there is a morphism t ∈ Hom_{C}(Y, X) satisfying s = s ∘ t ∘ s. (In the context of semigroups, this property is called regular or von Neumann regular.) The morphism t need not be unique, and also t need not be split itself, but one can always choose t to be split: just replace t by u = t ∘ s ∘ t; an easy calculation shows that we still have s = s ∘ u ∘ s and in addition u = u ∘ s ∘ u.

Lemma 2.1. Let C be a finite category and X, Y objects in C. Two idempotent endomorphisms e ∈ End_{C}(X) and f ∈ End_{C}(Y) are isomorphic if and only if e𝒥 f.

Proof. Suppose that e and f are isomorphic. That is, there are morphisms s ∈ f ∘Hom_{C}(X, Y)∘e and t ∈ e ∘ Hom_{C}(Y, X) ∘ f such that t ∘ s = e and s ∘ t = f. Thus e = t ∘ f ∘ s, hence Mor(C)∘e∘Mor(C) ⊆ Mor(C)∘f ∘Mor(C). Exchanging the roles of e and f yields the equality of these two sets, whence e𝒥 f. Conversely, suppose that Mor(C)∘e∘Mor(C) = Mor(C)∘f ∘Mor(C). Since e is an idempotent, this equality implies that e = t ∘ s for some s ∈ f ∘ Hom_{C}(X, Y) ∘ e and some t ∈ e ∘ Hom_{C}(Y, X) ∘ f. Similarly, f = s′ ∘ t′ for some s′ ∈ f ∘ Hom_{C}(X, Y) ∘ e and some t′ ∈ e ∘ Hom_{C}(Y, X) ∘ f. Thus t′ ∘ s and t ∘ s′ are both in e ∘ End_{C}(X) ∘ e and we have (t ∘ s′) ∘ (t′ ∘ s) = t ∘ (s′ ∘ t′) ∘ s = t ∘ f ∘ s = t ∘ s = e. Thus t ∘ s′ is left inverse to t′ ∘ s in e ∘ End_{C}(X) ∘ e. Since End_{C}(X) is finite this implies that t ∘ s′ is also right inverse to t′ ∘ s, thus t′ ∘ s ∘ t ∘ s′ = e. Setting u = s′ ∘ t′ ∘ s and v = t ∘ s′ ∘ t′, a straightforward verification yields v ∘ u = e and u ∘ v = f, whence e, f are isomorphic.

Lemma 2.2. Let C be a finite category and let s, t be morphisms in C such that t ∘ s is defined.

(i) We have s𝒥 (t ∘ s) if and only if s ℒ (t ∘ s). (ii) We have t𝒥 (t ∘ s) if and only if t ℛ (t ∘ s).

Proof. Suppose that s𝒥 (t∘s). Then s = u∘t∘s∘v for some morphisms u, v. Note that u∘t and v are endomorphisms of the target and source of s, respectively. Iterating the previous equation yields s = (u ∘ t)^{n} ∘ s ∘ v^{n} for all positive integers n. Since Mor(C) is finite, there is a positive integer n such that (u∘t)^{n} is an idempotent endomorphism. Thus s = (u∘t)^{n} ∘(u∘t)^{n} ∘s∘v^{n} = (u ∘ t)^{n} ∘ s, which is contained in Mor(C) ∘ t ∘ s, whence s ℒ (t ∘ s). The converse implication in (i) is obvious. Statement (ii) is proved analogously.

A semigroup satisfying the conclusions of 2.2 is called stable, but we do not use this terminology here in order to avoid confusion with the notion of stable module categories.

Lemma 2.3. Let C be a finite category and let s, t ∈ Mor(C) such that s𝒥 t. There is a morphism u ∈ Mor(C) such that the compositions u∘s is defined and such that we have s ℒ (u∘s) and (u ∘ s) ℛ t.

Proof. Since s𝒥 t there are morphisms u, v such that t = u∘s∘v. Since s𝒥 t we have s𝒥 (u∘s), hence s ℒ (u ∘ s) by 2.2. Similarly, we have (u ∘ s)𝒥 (u ∘ s ∘ v), hence (u ∘ s) ℛ (u ∘ s ∘ v).

Lemma 2.4. Let C be a small category and let s, t be morphisms in C. Suppose that s ℛ t or s ℒ t. Then s is split if and only if t is split.

Proof. Suppose that s is split; that is, s = s ∘ r ∘ s for some morphism r. If s ℒ t there are morphisms u, v such that s = u ∘ t and t = v ∘ s. Then t = v ∘ s = v ∘ s ∘ r ∘ s = t ∘ r ∘ s = t∘r ∘u∘t, and hence t is split. A similar argument shows that if s ℛ t then t is split. Exchanging the roles of s and t concludes the proof.

Lemma 2.5. Let C be a finite category and let J be a 𝒥 -equivalence class of morphisms in C. The following statements are equivalent.

(i) J contains an idempotent endomorphism of some object in C. (ii) J contains a split morphism in C. (iii) Every morphism in J is split. (iv) J ∘ J ∩ J is nonempty.

Proof. The implication (i) ⇒ (ii) is clear since every idempotent is split. In order to show (ii) ⇒ (iii), let s, t in J and assume that s is split. Let u, v ∈ Mor(C) be such that t = u ∘ s ∘ v. By 2.3 we have s ℒ (u ∘ s) ℛ t and by 2.4 both u ∘ s and t are split. In order to show the implication (iii)⇒ (iv) let s ∈ Mor(C) such that s ∘ t ∘ s = s for some t ∈ Mor(C). Then (s ∘ t)𝒥 s hence s ∘ t, s ∈ J and s = s ∘ t ∘ s ∈ J ∘ J ∩ J. For the implication (iv) ⇒ (i) let s ,t ∈ J such that s ∘ t is defined and in J. Then s𝒥 (s ∘ t) and t𝒥 (s ∘ t). By 2.2 we have s ℛ (s ∘ t) and t ℒ (s ∘ t), hence there are two morphisms u, v ∈ Mor(C) such that s = s ∘ t ∘ u and t = v ∘ s ∘ t. Note that t ∘ u ∈ J. Then t ∘ u ∘ t ∘ u = (v ∘ s ∘ t) ∘ u ∘ t ∘ u = v ∘ (s ∘ t ∘ u) ∘ t ∘ u = v ∘ s ∘ t ∘ u = t ∘ u.

A 𝒥 -class J in the morphism set Mor(C) of a finite category C is called split if J satisfies the equivalent conditions of Lemma 2.5. For e an idempotent endomorphism of an object X in C we denote as before by G_{e} the group of invertible elements of the monoid e ∘ End_{C}(X) ∘ e.

Lemma 2.6. Let C be a finite category, X an object in C and e an idempotent in End_{C}(X). Denote by J the 𝒥 -class of e in Mor(C). Then e ∘ End_{C}(X) ∘ e ∩ J = G_{e} and the complement of G_{e} in the monoid e ∘ End_{C}(X) ∘ e is a two-sided ideal.

Proof. Let s ∈ e ∘ End_{C}(X) ∘ e. Then s = e ∘ s, and hence, by 2.2, we have e𝒥 s if and only if e ℛ s. Since s = e∘s, the latter is equivalent to e = s∘u for some morphism u in e∘End_{C}(X)∘e, which is in turn equivalent to s ∈ G_{e} as the monoid e ∘ End_{C}(X) ∘ e is finite. The last statement is obvious.

The set of idempotents in the morphism set Mor(C) of a small category C is partially ordered via e ≤ f if e, f are idempotent endomorphisms of the same object in C such that e = e ∘ f = f ∘ e. The set of 𝒥 -classes in Mor(C) is partially ordered via J ≤ J′ if J, J′ are 𝒥 -classes such that for some (hence any) s ∈ J, s′ ∈ J′ we have Mor(C) ∘ s ∘ Mor(C) ⊆ Mor(C) ∘ s′ ∘ Mor(C). If J is the 𝒥 -class of a morphism s, then the set Mor(C) ∘ s ∘ Mor(C) is the disjoint union of all 𝒥 -classes J′ such that J′ ≤ J. The map sending an idempotent to its 𝒥 -class preserves these partial orders; more precisely:

Lemma 2.7. Let C be a finite category and let J, J′ be split 𝒥 -classes in Mor(C).

(i) If e ∈ J and e′ ∈ J′ are idempotents such that e ≤ e′ then J ≤ J′. (ii) If J ≤ J′ then for any idempotent e′ ∈ J′ there is an idempotent e ∈ J such that e ≤ e′.

Proof. If e ≤ e′ then e = e ∘ e′, hence e ∈ Mor(C) ∘ e′ ∘ Mor(C) which implies J ≤ J′, whence (i). Suppose that J ≤ J′. Then there are idempotents e ∈ J and e′ ∈ J′ such that Mor(C) ∘ e ∘ Mor(C) ⊆ Mor(C) ∘ e′ ∘ Mor(C). In particular, e = u ∘ e′ ∘ v for some morphisms u, v. Since e, e′ are idempotents we may choose u, v such that e ∘ u = u = u ∘ e′ and e′ ∘ v = v = v ∘ e. Then v ∘ u is defined and v ∘ u ∘ v ∘ u = v ∘ e ∘ u = v ∘ u, hence v ∘ u is an idempotent isomorphic to e. Moreover, we have e′ ∘ v ∘ u = v ∘ u = v ∘ u ∘ e′, hence v ∘ u ≤ e′, which proves (ii).

3 Proof of Theorem 1.1

Let C be a finite category. For any morphism s in Mor(C) we set

I_{s} = {t ∈ Mor(C) | s ∉ Mor(C) ∘ t ∘ Mor(C)} .

In other words, I_{s} is the union of all 𝒥 -classes J such that J_{s} ≰ J, where J_{s} is the 𝒥 -class containing s. In particular, I_{s} ∩ J_{s} = ∅. The set I_{s} is a two-sided ideal in Mor(C) and depends only on the 𝒥 -class of s. For any kC-module U we denote by Ann_{C}(U) the set of all morphisms in C which annihilate U. The following Proposition collects the main steps of the proof of Theorem 1.1.

Proposition 3.1. Let k be a commutative ring and C a finite category. Let S be a simple kC-module and J a minimal 𝒥 -class such that JS ≠ {0}. The following hold.

(i) J is split. (ii) J is unique. (iii) If e ∈ J, then Ann_{C}(S) = I_{e} (iv) If e ∈ J is an idempotent, then e ∘ End_{C}(X) ∘ e = G_{e} ∪ (e ∘ I_{e} ∘ e), and this union is disjoint.

Proof. (i) Let s ∈ J. As mentioned above, the set Mor(C) ∘ s ∘ Mor(C) is the disjoint union of all 𝒥 -classes J′ such that J′ ≤ J. Since any 𝒥 -class strictly smaller than J annihilates S, we have (kJ)S = k(Mor(C) ∘ s ∘ Mor(C))S. This is a nonzero submodule of the simple module S, hence equal to S. Applying this twice yields S = k(J ∘ J)S; in particular, J ∘ J does not annihilate S. Since J ∘ J is a subset of Mor(C) ∘ s ∘ Mor(C), it follows that J ∘ J must contain an element of J since otherwise J ∘ J would annihilate S. Thus, by 2.5, the 𝒥 -class J is split.

(ii) Let J′ be another 𝒥 -class such that J′S ≠ {0}. Then (kJ′)S = k(J′ ∘ J)S ≠ {0}. All elements in J′ ∘ J belong to 𝒥 -classes contained in J. Thus J′ ∘ J ∩ J ≠ ∅, which forces J ≤ J′.

(iii) Let s ∈ Ann_{C}(S). Since s annihilates S, every element of Mor(C)∘s∘Mor(C) annihilates S. Since J does not annihilate S, we have J ⊈ Mor(C) ∘ s ∘ Mor(C) and so s ∈ I_{e}. Conversely, if s ∈ I_{e} then e ∉ Mor(C) ∘ s ∘ Mor(C). Hence no 𝒥 -class J′ in Mor(C) ∘ s ∘ Mor(C) satisfies J ≤ J′. Since, by (ii), J is the unique minimal 𝒥 -class which does not annihilate S it follows that Mor(C) ∘ s ∘ Mor(C) annihilates S, whence s ∈ Ann_{C}(S).

(iv) As before we denote by J_{s} the 𝒥 -class containing a morphism s. Let s ∈ e∘End_{C}(X)∘e. If s ∈ J then s ∈ G_{e} by 2.6. Suppose s ∉ J. Then J_{s} < J_{e} = J, hence s annihilates S, by the minimality of J subject to not annihilating S. It follows that s ∈ Ann_{C}(S) = I_{e}, where the last equality uses (iii). Since e is an idempotent, this implies s ∈ e ∘ I_{e} ∘ e.

Proof of Theorem 1.1. Let S be a simple kC-module. Let e be a minimal idempotent endomorphism of an object X in C such that eS ≠ {0}. By [10, 6.2], eS is a simple k(e ∘ End_{C}(X) ∘ e)- module. By 2.7, the 𝒥 -class J of e is minimal such that JS ≠ {0}. Thus eS is annihilated by e ∘ I_{e} ∘ e, and hence, eS remains simple as a kG_{e}-module. Again by [10, 6.2], the module S is, up to isomorphism, determined by the isomorphism class of the pair (e, eS), and hence the map sending S to (e, eS) becomes injective on isomorphism classes. It becomes also surjective since any simple kG_{e}-module T can be viewed as a simple k(e ∘ End_{C}(X) ∘ e)-module in such a way that the complement of G_{e} in e ∘ End_{C}(X) ∘ e annihilates T.

4 Green relations for extensions of categories

Let C be a finite category and A an abelian group, written multiplicatively. Let α be a 2-cocycle of C with coefficients in A. That is, α is a map from pairs of composable morphisms in C to A satisfying the 2-cocycle identity. More precisely, if s, t are morphisms in C such that t ∘ s is defined, then α(t, s) is an element in A, with the property that if t, s, u are morphisms in C such that t ∘ s and u ∘ t are defined, then α(u ∘ t, s)α(u, t) = α(u, t ∘ s)α(t, s). The extension category of C by A associated with α is the category Ĉ with object set Ob(Ĉ) = Ob(C) and morphism set Mor(Ĉ) = Mor(C)×A, such that the composition in Ĉ is defined by (t, b)∘(s, a) = (t∘s, baα(t, s)), for any two morphisms s, t in C for which t ∘ s is defined and any a , b ∈ A. There is a canonical functor Ĉ → C which is the identity on objects and which sends a morphism (s, a) in Ĉ to the morphism s in C. The isomorphism class of the extension Ĉ of C by A depends only on the class of α in H^{2}(C; A). See [1], or also [23] for an introductory survey on extensions and cohomology of small categories, as well as further references. We will use in the sequel the elementary fact that if e, f are idempotent endomorphisms of objects X, Y, respectively, in C, then e, f are isomorphic if and only if there are morphisms s ∈ Hom_{C}(X, Y) and t ∈ Hom_{C}(Y, X) satisfying e = t∘s and f = s∘t. Indeed, if there are such morphisms s, t, then the morphisms s′ = f ∘s∘e and t′ = e ∘ t ∘ f are easily seen to still satisfy t′ ∘ s′ = e and s′ ∘ t′ = f, hence e and f are isomorphic. The converse is trivial. The following observation shows that equivalence classes of Green relations in Ĉ are precisely the inverse images of Green relations in C. We keep the notation above.

Lemma 4.1. Let s, t be morphisms in C and let a, b ∈ A.

(i) We have s ℒ t if and only if (s, a) ℒ (t, b). (ii) We have s ℛ t if and only if (s, a) ℛ (t, b). (iii) We have s𝒥 t if and only if (s, a)𝒥 (t, b). (iv) The morphism s is split if and only if (s, a) is split.

Proof. Suppose that s ℒ t. Then there is a morphism u such that s = u ∘ t. A short calculation shows that (u, aα(u, t)^{−1}b^{−1}) ∘ (t, b) = (s, a). Exchanging the roles of s and t yields that (s, a) ℒ (t, b). The converse implication in (i) is trivial. The statements (ii) and (iii) are proved similarly. For (iv), suppose that s = s∘u∘s for some morphism u in C. Setting b = a^{−1}α(s, u)α(s∘ u, s)^{−1} one checks that (s, a) = (s, a) ∘ (u, b) ∘ (s, a), hence (s, a) is split. The converse in (iv) is clear.

Note that 4.1 implies also that the Green relations in Ĉ satisfy the conclusion of 2.2 (that is, Ĉ is ‘stable’ in the semi-group sense, although Ĉ need no longer be finite because A need not be finite).

Lemma 4.2. Let X, Y be objects in C, let e ∈ End_{C}(X) and f ∈ End_{C}(Y) be idempotents and let s ∈ f ∘ Hom_{C}(X, Y) ∘ e. We have α(s, e) = α(e, e) and α(f, s) = α(f, f).

Proof. The 2-cocycle identity applied to s, e, e yields α(s ∘ e, e)α(s, e) = α(s, e ∘ e)α(e, e). Since s∘e = s and e∘e = e, the first equality follows from cancelling α(s, e) on both sides. The second equality follows similarly from the 2-cocycle identity applied to f, f, s.

Lemma 4.3. (i) The canonical functor Ĉ → C induces a bijection between the set of idempotents in Ĉ and the set of idempotents in C. The inverse of this bijection sends an idempotent e in C to the idempotent (e, α(e, e)^{−1}) in Ĉ. (ii) For any two idempotents e, f in C we have e ≤ f if and only if (e, α(e, e)^{−1}) ≤ (f, α(f, f)^{−1}). (iii) For any two idempotents e, f in C we have e ≅ f if and only if (e, α(e, e)^{−1}) ≅ (f, α(f, f)^{−1}).

Proof. Let e, f be idempotent endomorphisms of objects X, Y, respectively, in C. Let a ∈ A. Since (e, a) ∘ (e, a) = (e, a^{2}α(e, e)), the unique idempotent in the inverse image of e in the morphism set of Ĉ is (e, α(e, e)^{−1}), whence (i). Suppose that e ≤ f; that is, X = Y and e∘f = e = f ∘ e. Using the equality α(e, f) = α(f, f) from 4.2 we get that (e, α(e, e)^{−1}) ∘ (f, α(f, f)^{−1}) = (e, α(e, e)^{−1}). A similar argument yields (f, α(f, f)^{−1}) ∘ (e, α(e, e)^{−1}) = (e, α(e, e)^{−1}), hence (e, α(e, e)^{−1}) ≤ (f, α(f, f)^{−1}). The converse in (ii) is obvious. Suppose next that e ≅ f. Let s ∈ f ∘ Hom_{C}(X, Y) ∘ e and t ∈ e ∘ Hom_{C}(Y, X) ∘ f such that t ∘ s = e and s ∘ t = f. Set b = α(s, t)^{−1}α(f, f)^{−1}. Then (s, 1) ∘ (t, b) = (f, bα(s, t)) = (f, α(f, f)^{−1}). The 2-cocycle identity applied to s, t, s yields α(s, t ∘ s)α(t, s) = α(s ∘ t, s)α(s, t). Since t ∘ s = e we have α(s, t ∘ s) = α(s, e) = α(e, e), where the last equality uses 4.2. Similary, α(s ∘ t, s) = α(f, f). Thus α(e, e)α(t, s) = α(f, f)α(s, t). This equality implies that (t, b)∘(s, 1) = (e, α(e, e)^{−1}). This shows that (e, α(e, e)^{−1}) and (f, α(f, f)^{−1}) are isomorphic. The converse in (iii) is obvious.

5 Proof of Theorem 1.2

Let C be a finite category, k a commutative ring and α a 2-cocycle of C with coefficients in k^{×}. Denote by k_{α}C the corresponding twisted category algebra of C over k and by Ĉ the corresponding extension category of C by k^{×}. The map sending a morphism (s, λ) in Ĉ to λs induces a surjective algebra homomorphism kĈ → k_{α}C. For any subset M of Mor(C) denote by k_{α}M the k-submodule spanned by M in k_{α}C. For M, N two subsets of Mor(C) we have k_{α}M · k_{α}N = k_{α}(M ∘ N). In particular, if M is a 2-sided ideal in Mor(C) then k_{α}M is a 2-sided ideal in the algebra k_{α}C. As before, for any morphism s in C we denote by I_{s} the set of all morphisms t such that s ∉ Mor(C) ∘ t ∘ Mor(C) and for any k_{α}C-module U we denote by Ann_{C}(U) the set of all morphisms in C which annihilate U. By the previous remarks, Ann_{C}(U) is a 2-sided ideal in Mor(C). We keep the notation above. Proposition 3.1 carries over to twisted category algebras:

Proposition 5.1. Let S be a simple k_{α}C-module and J a minimal 𝒥 -class such that JS ≠ {0}. The following hold.

(i) J is split. (ii) J is unique. (iii) If e ∈ J, then Ann_{C}(S) = I_{e} (iv) If e ∈ J is an idempotent, then e ∘ End_{C}(X) ∘ e = G_{e} ∪ (e ∘ I_{e} ∘ e), and this union is disjoint.

Proof. Using the preceding remarks one sees that the proof of 3.1 carries over without any change. Alternatively, one can prove this by interpreting S as a simple module for the category algebra kĈ through the canonical homomorphism kĈ → k_{α}C, where Ĉ is the extension of C by k^{×} corresponding to α. Although Ĉ need not be finite, the conclusions of the results from §2 needed for the proof of 3.1 are easily seen to hold for Ĉ instead of C.

We need to verify that there is a well-defined notion of isomorphism classes of pairs (e, êS) as in the statement of 1.2. More precisely, we will show in the following two Propositions that isomorphic idempotents e, f in C give rise to an isomorphism of twisted group algebras k_{α}G_{e} ≅ k_{α}G_{f} which is uniquely determined up to an inner algebra automorphism.

Proposition 5.2. Let e, f be idempotent endomorphisms of objects X, Y, respectively, in C, let s ∈ f ∘ Hom_{C}(X, Y) ∘ e and t ∈ e ∘ Hom_{C}(Y, X) ∘ f satisfying t ∘ s = e and s ∘ t = f. For x ∈ G_{e} set β(x) = α(x, t)α(s, x ∘ t)α(e, e)^{−1}α(t, s)^{−1}. The map sending x ∈ G_{e} to β(x)(s ∘ x ∘ t) induces a k-algebra isomorphism k_{α}G_{e} ≅ k_{α}G_{f}.

Proof. Since the map sending x ∈ G_{e} to s ∘ x ∘ t is a group isomorphism G_{e} ≅ G_{f}, the map sending x ∈ G_{e} to β(x)(s ∘ x ∘ t) induces a k-linear isomorphism k_{α}G_{e} ≅ k_{α}G_{f}. It remains to show that this map is multiplicative. Let x, y ∈ G_{e}. The image of the product x · y = α(x, y)(x ∘ y) is equal to α(x, y)β(x ∘ y)(s ∘ x ∘ y ∘ t). The product of the images of x, y in k_{β}G_{f} is equal to α(s ∘ x ∘ t, s ∘ y ∘ t)β(x)β(y). Thus we need to show the equality

α(x, y)β(x ∘ y) = α(s ∘ x ∘ t, s ∘ y ∘ t)β(x)β(y)

We consider first the left side in this equation, which is equal to α(x, y)α(x ∘ y, t)α(s, x ∘ y ∘ t)α(e, e)^{−1}α(t, s)^{−1}. The 2-cocycle identity applied to x, y, t in the first two factors of this product yields the expression α(y, t)α(x ∘ y, t)α(s, x ∘ y ∘ t)α(e, e)^{−1}α(t, s)^{−1}. The 2-cocycle identity applied to s, x, y ∘ t in the second and third factor of this product yields the equality

α(x, y)β(x ∘ y) = α(y, t)α(s, x)α(s ∘ x, y ∘ t)α(e, e)^{−1}α(t, s)^{−1}

We need to show that this is equal to α(s∘x∘t, s∘y ∘t)β(x)β(y). The 2-cocycle identity applied to s∘x∘t, s, y ∘t yields α(s∘x∘t, s∘y ∘t) = α(s∘x, y ∘t)α(s∘x∘t, s)α(s, y ∘t)^{−1}. The 2-cocycle identity applied to s ∘ x, t, s in the middle term of this expression yields α(s ∘ x ∘ t, s ∘ y ∘ t) = α(s ∘ x, y ∘ t)α(s ∘ x, t ∘ s)α(t, s)α(s ∘ x, t)^{−1}α(s, y ∘ t)^{−1}. Using t ∘ s = e and α(s ∘ x, e) = α(e, e) from 4.2 yields α(s ∘ x ∘ t, s ∘ y ∘ t) = α(s ∘ x, y ∘ t)α(e, e)α(t, s)α(s ∘ x, t)^{−1}α(s, y ∘ t)^{−1}. Multiplying this equation by β(x)β(y) yields α(s ∘ x ∘ t, s ∘ y ∘ t)β(x)β(y) = α(s ∘ x, y ∘ t)α(s ∘ x, t)^{−1}α(x, t)α(s, x ∘ t)α(e, e)^{−1}α(t, s)^{−1}α(y, t). The 2-cocycle identity applied to s, x, t in the second, third and fourth factor shows that this is indeed equal to α(x, y)β(x∘y) as required.

Lemma 5.3. Let e, f be idempotent endomorphisms of objects X, Y, respectively, in C, let s ∈ f ∘ Hom_{C}(X, Y) ∘ e and t ∈ e ∘ Hom_{C}(Y, X) ∘ f satisfying t ∘ s = e and s ∘ t = f. We have α(t, s)α(e, e) = α(s, t)α(f, f). In particular, if e = f then α(t, s) = α(s, t).

Proof. This follows from 4.2 and the 2-cocycle identity applied to the morphisms t, s and t.

Proposition 5.4. Let e, f be idempotent endomorphisms of objects X, Y, respectively, in C. Let s, s′ ∈ f ∘Hom_{C}(X, Y)∘e and t, t′ ∈ e∘Hom_{C}(Y, X)∘f satisfying t∘s = e = t′ ∘s′ and s∘t = f = s′ ∘ t′. For x ∈ G_{e} set β(x) = α(x, t)α(s, x ∘ t)α(e, e)^{−1}α(t, s)^{−1}. For y ∈ G_{f} set β′(y) = α(y, s′)α(t′, y∘s′)α(f, f)^{−1}α(s′, t′)^{−1}. The map sending x ∈ G_{e} to β(x)β′(s∘x∘t)(t′ ∘s∘x∘t∘s′) induces an inner automorphism of the k-algebra k_{α}G_{e}.

Proof. We will show that the map sending x ∈ G_{e} to β(x)β′(s ∘ x ∘ t)(t′ ∘ s ∘ x ∘ t ∘ s′) coincides with conjugation by a = t′ ∘ s in k_{α}G_{e}. Note that t ∘ s′ is the inverse of t′ ∘ s in the group G_{e} but not necessarily in the algebra k_{α}G_{e}. More precisely, since the unit element in k_{α}G_{e} is equal to α(e, e)^{−1}e, the inverse of a = t′ ∘ s in k_{α}G_{e} is equal to a^{−1} = α(e, e)^{−1}α(t ∘ s′, t′ ∘ s)^{−1}(t ∘ s′). We have

a · x · a^{−1} = α(e, e)^{−1}α(t ∘ s′, t′ ∘ s)^{−1}α(t′ ∘ s, x)α(t′ ∘ s ∘ x, t ∘ s′)(t′ ∘ s ∘ x ∘ t ∘ s′)

Thus we need to show the equality

β(x)β′(s ∘ x ∘ t) = α(e, e)^{−1}α(t ∘ s′, t′ ∘ s)^{−1}α(t′ ∘ s, x)α(t′ ∘ s ∘ x, t ∘ s′)

Bringing all terms with an inverse sign to the right side and cancelling one factor α(e, e) shows that this equality is equivalent to the equality

α(x, t)α(s, x ∘ t)α(s ∘ x ∘ t, s′)α(t′, s ∘ x ∘ t ∘ s′)α(t′ ∘ s, t ∘ s′) =
α(t, s)α(f, f)α(s′, t′)α(t′ ∘ s, x)α(t′ ∘ s ∘ x, t ∘ s′)

The 2-cocycle identity for s, x ∘ t, s′ applied to the second an third factor on the left side shows that the left side is equal to

α(x, t)α(x ∘ t, s′)α(s, x ∘ t ∘ s′)α(t′, s ∘ x ∘ t ∘ s′)α(t′ ∘ s, t ∘ s′)

The 2-cocycle identity for x, t, s′ applied to the first and second factor, and the 2-cocycle identity for t′, s, x ∘ t ∘ s′ applied to the third and fourth factor of this expression show that the left side above is equal to

α(t, s′)α(x, t ∘ s′)α(t′, s)α(t′ ∘ s, x ∘ t ∘ s′)α(t′ ∘ s, t ∘ s′)

The 2-cocycle identity for t′, s, t ∘ s′ applied to the third and fifth factor yields the expression

α(t, s′)α(x, t ∘ s′)α(t′s′)α(s, t ∘ s′)α(t′ ∘ s, x ∘ t ∘ s′)

The 2-cocycle identity for s, t, s′ applied to the first and fourth factor (using s ∘ t = f and α(f, s′) = α(f, f) from 4.2) yields

α(f, f)α(s, t)α(x, t ∘ s′)α(t′, s′)α(t′ ∘ s, x ∘ t ∘ s′)

The 2-cocycle identity for t∘s′, x, t∘s′ applied to the third and fifth factor yields the expression

α(f, f)α(s, t)α(t′ ∘ s, x)α(t′ ∘ s ∘ x, t ∘ s′)α(t′, s′)

Using 5.3 twice shows that this is equal to the expression on the right side of the above equation.

Proof of Theorem 1.2. The above results in this section imply that the proof of 1.1 carries over to twisted category algebras. Let S be a simple k_{α}C-module and e a minimal idempotent endomorphism of an object X in C such that êS ≠ {0}, where ê = α(e, e)^{−1}e. By [10, 6.2], êS is a simple k_{α}(e ∘ End_{C} ∘ e)-module. By 2.7, the 𝒥 -class J of e is minimal such that JS ≠ {0}. Thus êS is annihilated by e ∘ I_{e} ∘ e, and hence, êS remains simple as a k_{α}G_{e}-module. By 5.2 and 5.4, we have a well-defined notion of isomorphism between pairs if the form (e, êS). Again by [10, 6.2], the module S is, up to isomorphism, determined by the isomorphism class of the pair (e, êS), and hence the map sending S to (e, êS) becomes injective on isomorphism classes. It becomes also surjective since, by 5.1, any simple k_{α}G_{e}-module T can be viewed as a simple k_{α}(e ∘ End_{C}(X) ∘ e)-module in such a way that the complement of G_{e} in e ∘ End_{C}(X) ∘ e annihilates T.

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