A $q$-analogue of derivations on the tensor algebra and the $q$-Schur-Weyl duality

TL;DR

This paper introduces a $q$-analogue of an extended tensor algebra that unifies tensor and Iwahori-Hecke algebras, and uses it to prove the $q$-Schur-Weyl duality through natural derivations.

Contribution

It develops a new $q$-analogue algebra that combines tensor and Iwahori-Hecke algebras and provides a straightforward proof of the $q$-Schur-Weyl duality.

Findings

Construction of a $q$-analogue algebra containing tensor and Iwahori-Hecke algebras

Definition of natural derivations on this algebra

Simplified proof of the $q$-Schur-Weyl duality

Abstract

This paper presents a $q$-analogue of an extension of the tensor algebra given by the same author. This new algebra naturally contains the ordinary tensor algebra and the Iwahori-Hecke algebra type $A$ of infinite degree. Namely this algebra can be regarded as a natural mix of these two algebras. Moreover, we can consider natural "derivations" on this algebra. Using these derivations, we can easily prove the $q$-Schur-Weyl duality (the duality between the quantum enveloping algebra of the general linear Lie algebra and the Iwahori-Hecke algebra of type $A$).