A categorical approach to classical and quantum Schur-Weyl duality

TL;DR

This paper employs category theory to unify classical and quantum Schur-Weyl dualities, providing a framework that connects various algebraic structures and their representations through monoidal categories and functors.

Contribution

It introduces a categorical framework that unifies classical and quantum Schur-Weyl dualities, including constructions of functors and actions on braided categories.

Findings

Unified categorical approach to Schur-Weyl duality

Construction of monoidal categories from algebra sequences

Universal properties leading to functorial connections

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.