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

Abstract

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.