A categorical approach to classical and quantum Schur-Weyl duality

Abstract

We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras exemplified by the sequence of group algebras of the symmetric groups and use them to introduce associated monoidal categories. Universal properties of these categories lead to uniform constructions of the Drinfeld functor connecting representation theories of the degenerate affine Hecke algebras and the Yangians and of its q-analogue. Moreover, we construct actions of these categories on certain (infinitesimal) braided categories containing a Hecke object.