Mixed Tensor Representations and Rational Representations for the General Linear Lie Superalgebras

Shader, Chanyoung Lee; Moon, Dongho

doi:10.1081/agb-120013185

Cited by 18 publications

(16 citation statements)

References 8 publications

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“…Let − → B r,s be the superspace with a basis consisting of (r, s)-superdiagrams. The (r, s)superdiagrams without marked edges form a basis of the walled Brauer algebra B r,s (δ) (δ ∈ C) (See, for example, [1,3,4,9,14]).…”

Section: The Walled Brauer Superalgebrasmentioning

confidence: 99%

Mixed Schur–Weyl–Sergeev duality for queer Lie superalgebras

Jung

Kang

2014

Journal of Algebra

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We introduce a new family of superalgebras − → B r,s for r, s ≥ 0 such that r + s > 0, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let q(n) be the queer Lie superalgebra, V = C n|n the natural representation of q(n) and W the dual of V. We prove that, if n ≥ r + s, the superalgebra − → B r,s is isomorphic to the supercentralizer algebra End q(n) (V ⊗r ⊗W ⊗s ) op of the q(n)-action on the mixed tensor space V ⊗r ⊗W ⊗s . As an ingredient for the proof of our main result, we construct a new diagrammatic realization − → D k of the Sergeev superalgebra Ser k . Finally, we give a presentation of − → B r,s in terms of generators and relations.2000 Mathematics Subject Classification. 17B10, 05E10. Key words and phrases. mixed Schur-Weyl-Sergeev duality, queer Lie superalgebras, walled Brauer superalgebras.

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Section: The Walled Brauer Superalgebrasmentioning

confidence: 99%

Mixed Schur–Weyl–Sergeev duality for queer Lie superalgebras

Jung

Kang

2014

Journal of Algebra

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“…In this section, we collect some preliminary results from [2] and [13] that we needed for the development of the paper.…”

Section: Preliminariesmentioning

confidence: 99%

“…For complex general linear Lie superalgebra g = gl(m, n), Shader and Moon [13] studied the mixed tensor representation (V * ) ⊗r ⊗ V ⊗s of gl(m, n), where V = C m+n is the natural representation of gl(m, n) and V * is the dual of V and they proved that centralizer algebra End g ((V * ) ⊗r ⊗ V ⊗s ) is isomorphic to the walled Brauer algebra B r,s (m − n) for r + s ≤ m − n. Moreover, Brundan and Stroppel [2], also studied the mixed tensor representation V ⊗r ⊗ (V * ) ⊗s of g = gl(m, n) and proved that the centralizer algebra End g (V ⊗r ⊗ (V * ) ⊗s ) op is isomorphic to the walled Brauer algebra B r,s (m−n) when r+s < (m+1)(n+1).…”

Section: Introductionmentioning

confidence: 99%

Walled Signed Brauer Algebras as Centralizer Algebras of the Mixed Tensor Representation of Lie Superalgebras

Kethesan¹

2015

Int. J. of Pure and Appl. Math.

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In this paper, we prove that the walled signed Brauer algebra − → D r,s (x) is embedded in a canonical fashion as a subalgebra of a walled Brauer algebra B 2r,2s (x) and the walled signed Brauer algebra − → D r,s (x) is the centralizer algebra of σ = (1 2)(3 4) . . . (2r−1 2r)(2r+1 2r+2) . . . (2r+2s−1 2r+2s) in the walled Brauer algebra B 2r,2s (x). Finally we prove that, if 2r+2s < (m+1)(n+1), the walled signed Brauer algebra − → D r,s (δ) is isomorphic to the centralizer algebra End g 2r+2s (V ⊗r ⊗ (V * ) ⊗s ) op of the g 2r+2s -action on the mixed tensor space (V ⊗r ⊗ (V * ) ⊗s ); where δ = m − n, g = gl(m, n) is the complex general linear Lie superalgebra, V = W ⊗ W, W = C m|n is the natural representation of g and V * is the dual of V.

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“…By replacing gl(n) by the quantum enveloping algebra U q (gl(n)) and V = C n by V q = C(q) n , we obtain as the centralizer algebra the quantum walled Brauer algebra studied in [3,4,7,11,14]. Super versions of the above constructions have been investigated with the following substitutions: Replace gl(n) by gl(m|n), C n by C (m|n) ; U q (gl(n)) by U q (gl(m|n)); and C(q) n by C(q) (m|n) as in [15,16].…”

Section: Introductionmentioning

confidence: 99%

Quantum walled Brauer–Clifford superalgebras

et al. 2016

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Abstract. We introduce a new family of superalgebras, the quantum walled Brauer-Clifford superalgebras BCr,s(q). The superalgebra BCr,s(q) is a quantum deformation of the walled BrauerClifford superalgebra BCr,s and a super version of the quantum walled Brauer algebra. We prove that BCr,s(q) is the centralizer superalgebra of the action of Uq(q(n)) on the mixed tensor space V r,s q⊗s when n ≥ r + s, where Vq = C(q) (n|n) is the natural representation of the quantum enveloping superalgebra Uq(q(n)) and V * q is its dual space. We also provide a diagrammatic realization of BCr,s(q) as the (r, s)-bead tangle algebra BTr,s(q). Finally, we define the notion of q-Schur superalgebras of type Q and establish their basic properties.

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Mixed Tensor Representations and Rational Representations for the General Linear Lie Superalgebras

Cited by 18 publications

References 8 publications

Mixed Schur–Weyl–Sergeev duality for queer Lie superalgebras

Mixed Schur–Weyl–Sergeev duality for queer Lie superalgebras

Walled Signed Brauer Algebras as Centralizer Algebras of the Mixed Tensor Representation of Lie Superalgebras

Quantum walled Brauer–Clifford superalgebras

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