The Lusztig automorphism of $U_q(\mathfrak{sl}_2)$ from the equitable point of view

TL;DR

This paper explores the quantum algebra $U_q(\mathfrak{sl}_2)$ using the equitable presentation to describe the Lusztig automorphism and operator, providing a new perspective on their structure.

Contribution

It introduces an equitable presentation of $U_q(\mathfrak{sl}_2)$ and characterizes the Lusztig automorphism within this framework, offering novel insights.

Findings

Explicit description of the Lusztig automorphism in equitable form

New perspective on the structure of $U_q(\mathfrak{sl}_2)$ automorphisms

Enhanced understanding of the Lusztig operator from the equitable viewpoint

Abstract

We consider the quantum algebra $U_q(\mathfrak{sl}_2)$ in the equitable presentation. From this point of view, we describe the Lusztig automorphism and the corresponding Lusztig operator.