Presenting quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl(2)

TL;DR

This paper provides a presentation of quantum Schur algebras as quotients of the quantized universal enveloping algebra of gl(2), linking their structures and bases explicitly.

Contribution

It introduces a generator-and-relations presentation of quantum Schur algebras compatible with the gl(2) quantum enveloping algebra, and identifies their integral form and basis.

Findings

Presented quantum Schur algebras via generators and relations.

Located the integral form within the algebra and related it to Lusztig's basis.

Established compatibility with the quantum enveloping algebra of gl(2).

Abstract

We obtain a presentation of quantum Schur algebras (over the field Q(v)) by generators and relations. This presentation is compatible with the usual presentation of the quantized universal enveloping algebra of the Lie algebra gl(2). We also locate the ``integral'' form of the quantum Schur algebra within the presented algebra and show it has a basis which is closely related to Lusztig's basis of the integral form of the quantized enveloping algebra.