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

Minoru Itoh

Published 2014-11-15Version 1

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 $A$).