Non-vanishing Gram determinants for cyclotomic Nazarov–Wenzl and Birman–Murakami–Wenzl algebras

Rui, Hebing; Si, Mei

doi:10.1016/j.jalgebra.2011.03.016

Cited by 7 publications

(5 citation statements)

References 36 publications

(102 reference statements)

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“…Finally, we remark that Theorem 2.18 can be proved by similar arguments for B n . In order to give the detailed proof, we need results which are similar to Theorem 3.3, Definition 3.5, Theorem 3.6 and Lemma 4.3 etc, which can found in [3,7,12,13,16] etc. We leave the details to the reader.…”

Section: The Main Resultsmentioning

confidence: 99%

Singular parameters for the Birman–Murakami–Wenzl algebra

Rui

2012

Journal of Pure and Applied Algebra

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Section: The Main Resultsmentioning

confidence: 99%

Singular parameters for the Birman–Murakami–Wenzl algebra

Rui

2012

Journal of Pure and Applied Algebra

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“…In the non-semisimple case, blocks of A are determined. In particular, when the (degenerate) cyclotomic Hecke algebras are semisimple and char k = 2, we use results on the classification of blocks for cyclotomic Nazarov-Wenzl algebras and cyclotomic Birman-Murakami-Wenzl algebras in [40,41] to classify blocks of A completely. In this case, A-lfdmod is an upper finite highest weight category in the sense of [18].…”

Section: Introductionmentioning

confidence: 99%

“…Note that the cell modules 1 k ∆(ν)'s coincide with cell modules of A k studied in [38] and [46] (see Remark 6.9). By [38,Theorem 3.12] When the Hecke algebra (resp., (degenerate) cyclotomic Hecke algebra) is semisimple over k (with characteristic not two), a necessary and sufficient condition on cell-link for the Birman-Murakami-Wenzl algebra (resp., the cyclotomic Nazarov-Wenzl algebra and the cyclotomic Birman-Murakami-Wenzl algebra) is given in [41,Propositions 4.6,4.10] (resp., [40,Propositions 5.10,5.21]). Using Proposition 6.10 yields an explicit combinatorial description on the blocks of A.…”

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confidence: 99%

Representations of weakly triangular categories

Gao¹,

Rui²,

Song³

2020

Preprint

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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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“…Such algebras have been studied extensively by many authors in [8][9][10][17][18][19][21][22][23][24], etc. By adding some additional conditions on the parameters of B r,n , one gets a well behaved representation theory of cyclotomic BMW algebras.…”

Section: Introductionmentioning

confidence: 99%

Morita equivalence for cyclotomic BMW algebras

2015

Journal of Algebra

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In this paper, we determine explicitly the singular parameters for cyclotomic BMW algebras B r,n over an arbitrary field. Equivalently, we give a criterion for B r,n being Morita equivalent to the direct sum of some Ariki-Koike algebras. This generalizes König and Xi's work on Brauer algebras [14, Theorem 7.3] and our previous work on Brauer and BirmanMurakami-Wenzl algebras [20, Proposition 2.15].

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Non-vanishing Gram determinants for cyclotomic Nazarov–Wenzl and Birman–Murakami–Wenzl algebras

Cited by 7 publications

References 36 publications

Singular parameters for the Birman–Murakami–Wenzl algebra

Singular parameters for the Birman–Murakami–Wenzl algebra

Representations of weakly triangular categories

Morita equivalence for cyclotomic BMW algebras

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