Quasi-hereditary twisted category algebras

Linckelmann, Markus; Stolorz, Michał

doi:10.1016/j.jalgebra.2013.02.036

Cited by 4 publications

(6 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, we suppose that the category C is split, that is, every morphism in C is a split morphism, see 3.1(a). It has been shown by the authors in [2] and, independently, by Linckelmann and Stolorz in [18] that the resulting twisted category algebra k α C is quasi-hereditary in the sense of [6]. In [2] we also determined the standard modules of k α C with respect to a natural partial order ≤ on the labelling set Λ of isomorphism classes of simple modules, which depends only on the first entries of pairs in Λ and is explained in 3.6.…”

Section: Introductionmentioning

confidence: 74%

“…It is known, by work of Wilcox [27], that diagram algebras such as Brauer algebras, Temperley-Lieb algebras, partition algebras, and relatives of these arise naturally as twisted split category algebras and twisted regular monoid algebras; for a list of references, see the introductions to [2,21]. Our initial motivation for studying the structure of twisted category algebras comes from our results in [1], where we have shown that the double Burnside algebra of a finite group over k is isomorphic to a k-algebra that is obtained from a twisted split category algebra by idempotent condensation.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quasi-hereditary structure of twisted split category algebras revisited

Boltje

Danz

2015

Journal of Algebra

View full text Add to dashboard Cite

Available online xxxx Communicated by Michel Broué MSC: 16G10 20M17 19A22 Keywords: Split category Regular monoid Quasi-hereditary algebra Highest weight category Biset functor Brauer algebraLet k be a field of characteristic 0, let C be a finite split category, let α be a 2-cocycle of C with values in the multiplicative group of k, and consider the resulting twisted category algebra A := k α C. Several interesting algebras arise that way, for instance, the Brauer algebra. Moreover, the category of biset functors over k is equivalent to a module category over a condensed algebra εAε, for an idempotent ε of A. In [2] the authors proved that A is quasi-hereditary (with respect to an explicit partial order on the set of irreducible modules), and standard modules were given explicitly. Here, we improve the partial order by introducing a coarser order P leading to the same results on A, but which allows to pass the quasi-heredity result to the condensed algebra εAε describing biset functors, thereby giving a different proof of a quasi-heredity result of Webb, see [21]. The new partial order P has not been considered before, even in the special cases, and we evaluate it explicitly for the case of biset functors and the Brauer algebra. It also puts further restrictions on the possible composition factors of standard modules.

show abstract

Section: Introductionmentioning

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Quasi-hereditary structure of twisted split category algebras revisited

Boltje

Danz

2015

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…Following [8] and [9], given a finite category C, the isomorphism classes of simple kC-modules are parametrised by isomorphism classes of pairs (e, T ), with e an idempotent endomorphism of some object X in C and T a simple kG e -module, where G e is the group of all invertible elements in the monoid e • End C (X) • e. Such a pair (e, T ) is called a weight if the simple kG e -module T is in addition projective. The associated weight algebra W (kC) is an algebra of the form W (kC) = c · kC · c for some idempotent c which acts as the identity on all simple kC-modules parametrised by a weight and which annihilates all simple modules not parametrised by a weight; we review this in §2 below.…”

Section: Introductionmentioning

confidence: 99%

A version of Alperinʼs weight conjecture for finite category algebras

Linckelmann

2014

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

Citation: Linckelmann, M. (2013). A version of alperin's weight conjecture for finite category algebras. Journal of Algebra, 398, pp. 386-395. doi: 10.1016Algebra, 398, pp. 386-395. doi: 10. /j.jalgebra.2013 This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent Markus LinckelmannAbstract Let p be a prime and k an algebraically closed field of characteristic p. We construct a functor C → OC on the category of finite categories with the property that if G = C is a finite group, then OC is the orbit category of p-subgroups of G. This leads to an extension of Alperin's weight conjecture to any finite category C, stating that the number of isomorphism classes of simple kC-modules should be equal to that of the weight algebra W (kOC) of OC. We show that the versions of Alperin's weight conjecture for finite groups and for finite categories are in fact equivalent.

show abstract

“…Proofs of the quasi-heredity of the aforementioned diagram algebras can, for instance, be found in [9,15,17,18,20,22], and also in work of König-Xi [12], who established necessary and sufficient criteria for a cellular algebra to be quasi-hereditary. For a more detailed discussion of the history of proofs that Brauer algebras, Temperley-Lieb algebras, and partition algebras are quasi-hereditary over coefficient fields of characteristic 0, we refer to [14].…”

mentioning

confidence: 99%

“…We recently learnt that Linckelmann and Stolorz, see [14,Theorem 1.1], independently proved that, under certain conditions on the category, finite twisted category algebras are quasihereditary in characteristic 0. These conditions on the category are even weaker than being split, and therefore the results in [14] imply Theorem 3.5. However, the two approaches are slightly different; for instance, we explicitly determine the radical of the twisted category algebra as part of our proof.…”

mentioning

confidence: 99%

Twisted split category algebras as quasi-hereditary algebras

Boltje

Danz

2012

Arch. Math.

View full text Add to dashboard Cite

We show that if C is a finite split category, k is a field of characteristic 0 and α is a 2-cocycle of C with values in k × then the twisted category algebra k α C is quasi-hereditary. IntroductionThroughout this paper we assume that C is a finite category, that is, the objects of C form a finite set, and for every X, Y ∈ Ob(C), the morphism set Hom C (X, Y ) is finite. The category C is called split if, for each morphism s ∈ Hom C (X, Y ), there is a (not necessarily unique) morphism t ∈ Hom C (Y, X) such that s • t • s = s. Note that u := t • s • t then also satisfies s • u • s = s, and also u • s • u = u. In the special case where C has only one object this leads to the notion of a regular monoid, see [10].Let k be a field, and let α be a 2-cocycle of C with values in k × . That is, for every pair s, t ∈ Mor(C) such that t •s exists, one has an element α(t, s) ∈ k × such that the following holds: for any s, t, u ∈ Mor(C) such that t • s and u • t exist, one has α(u • t, s)α(u, t) = α(u, t • s)α(t, s). We will study the twisted category algebra k α C, that is, the k-vector space with basis Mor(C) and multiplicationThe aim of this paper, see Theorem 3.5, is to show that if C is a finite split category and if k has characteristic 0 then k α C is a quasi-hereditary algebra. This generalizes a result of Putcha, see [16], who proved that regular monoid algebras are quasi-hereditary over k = C. In Theorem 4.2 we identify the standard modules, generalizing Putcha's results in [16].

show abstract

Quasi-hereditary twisted category algebras

Cited by 4 publications

References 16 publications

Quasi-hereditary structure of twisted split category algebras revisited

Quasi-hereditary structure of twisted split category algebras revisited

A version of Alperinʼs weight conjecture for finite category algebras

Twisted split category algebras as quasi-hereditary algebras

Contact Info

Product

Resources

About