Schur–Weyl quasi-duality and (co)triangular Hopf quasigroups

Schur–Weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic and plays a central role in many parts of algebraic Lie theory. In this paper, we will introduce the notion of Schur–Weyl quasi-duality and study it. For this, generally, we consi...

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Bibliographic Details
Published in:Journal of mathematical physics 2020-05, Vol.61 (5)
Main Authors: Shi, Guodong, Wang, Shuanhong
Format: Article
Language:English
Subjects:
Algebra
Braiding
Lie groups
Physics
Vector space
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Summary:Schur–Weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic and plays a central role in many parts of algebraic Lie theory. In this paper, we will introduce the notion of Schur–Weyl quasi-duality and study it. For this, generally, we consider a braided vector space (V,c) and its braided Lie algebra Endk(V)(−). Then, we can construct its braided enveloping algebra U(Endk(V)(−)), which is a connected braided c-cocommutative Hopf algebra. Let H be a triangular Hopf quasigroup with bijective antipode and B be a cotriangular Hopf quasigroup with bijective antipode. Let V be any finite dimensional vector space in the category LQ(H,R)(B,σ) of generalized Long quasimodules. We show that (U((EndkV)(−))⋆H⋆B,kSn,V⊗n) is a Schur–Weyl quasi-duality under suitable conditions.
ISSN:0022-2488
1089-7658
DOI:10.1063/5.0005803