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 consider a braided vector space (V,c) and its braided Lie algebra End(k)(V)((-)). Then, we can construct its braided enveloping algebra U(End(k)(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 <mml:mmultiscripts>LQ(H,R)(B,sigma)</mml:mmultiscripts> of generalized Long quasimodules. We show that (U((EndkV)(-))star H star B,kSn,V circle times n) is a Schur-Weyl quasi-duality under suitable conditions.