Representations of Brauer category and categorification

Rui, Hebing; Song, Linliang

doi:10.1016/j.jalgebra.2020.04.013

Cited by 7 publications

(5 citation statements)

References 27 publications

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“…where p is the characteristic of k. Let A be the algebra associated to CB(u). When CB(u) is the Brauer category, i.e., m = 1, thanks to [45], A admits an upper finite triangular decomposition in the sense of [18]. Hence A admits an upper finite weakly triangular decomposition in the sense of Definition 2.…”

Section: (Degenerate) Cyclotomic Hecke Algebras Givenmentioning

confidence: 99%

“…Then the stratification of the upper finite stratified category A-lfdmod in Theorem 3.10 can be refined such that ρ is replaced by the weight function wt : Λ → P with the order on P given in (5.28). See [45] for Brauer category. In the current case, the result follows from arguments similar to those for Brauer category in [45].…”

Section: Categorical Actions Onmentioning

confidence: 99%

“…See [45] for Brauer category. In the current case, the result follows from arguments similar to those for Brauer category in [45]. Of course, we need the fact that wt(µ) ≺ wt(λ) if [ ∆(λ) : L(µ)] = 0 and λ = µ.…”

Section: Categorical Actions Onmentioning

confidence: 99%

“…Many k-linear categories come into their theory. Archetypal examples include the Brauer category [31,45], the oriented Brauer category [14], the oriented skein category [13] and those triangular categories in [47]. All locally unital algebras associated to these categories admit upper finite triangular decompositions.…”

Section: Introductionmentioning

confidence: 99%

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Representations of weakly triangular categories

Gao¹,

Rui²,

Song³

2020

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A new class of locally unital and locally finite dimensional algebras A over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of an upper finite triangular decomposition. Any locally unital algebra which admits an upper finite Cartan decomposition is Morita equivalent to some special locally unital algebra A which admits an upper finite weakly triangular decomposition. It is established that the category A-lfdmod of locally finite dimensional left A-modules is an upper finite fully stratified category in the sense of Brundan-Stroppel. Moreover, A is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple. Furthermore, certain endofunctors are defined and give categorical actions of some Lie algebras on the subcategory of A-lfdmod consisting of all objects which have a finite standard filtration. In the case A is the locally unital algebra associated to one of cyclotomic oriented Brauer categories, cyclotomic Brauer categories and cyclotomic Kauffman categories, A admits an upper finite weakly triangular decomposition. This leads to categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of sl∞ or ŝle. Finally, we study representations of A associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisimplicity of A over an arbitrary field, and on A-lfdmod being upper finite highest weight category in the sense of Brundan-Stroppel, and on Morita equivalence between A and direct sum of infinitely many (degenerate) cyclotomic Hecke algebras.

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Section: (Degenerate) Cyclotomic Hecke Algebras Givenmentioning

confidence: 99%

Section: Categorical Actions Onmentioning

confidence: 99%

Section: Categorical Actions Onmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Representations of weakly triangular categories

Gao¹,

Rui²,

Song³

2020

Preprint

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“…For the Brauer category, further results have been obtained. More explicitly, using the representation theory of Brauer category, they [34] gave a categorification related to the Brauer algebra in [6]. In this picture, certain modules of coideal algebras in [3] come into picture.…”

Section: Introductionmentioning

confidence: 99%

A basis theorem for the affine Kauffmann category and its cyclotomic quotients

Gao,

Rui,

Song

2020

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The affine Kauffmann category is a strict monoidal category and can be considered as a q-analogue of the affine Brauer category in (Rui et al. in Math. Zeit. 293, 503-550, 2019). In this paper, we prove a basis theorem for the morphism spaces in the affine Kauffmann category. The cyclotomic Kauffmann category is a quotient category of the affine Kauffmann category. We also prove that any morphism space in this category is free over an integral domain K with maximal rank if and only if the u-admissible condition holds in the sense of Definition 1.13.

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