The Laurent Extension of Quantum Plane: a Complete List of $U_q(\mathfrak{sl}_2)$-Symmetries

TL;DR

This paper completes the classification of all $U_q(rak{sl}_2)$-symmetries on the Laurent extension of the quantum plane, providing a comprehensive list including both generic and non-generic cases.

Contribution

It provides the first complete classification of $U_q(rak{sl}_2)$-symmetries on the Laurent quantum plane, extending previous partial results.

Findings

Complete list of $U_q(\mathfrak{sl}_2)$-symmetries on the Laurent quantum plane

Reproduction of partial results for non-trivial Cartan actions

Introduction of non-generic symmetries

Abstract

This work finishes a classification of $U_q(\mathfrak{sl}_2)$-symmetries on the Laurent extension $\mathbb{C}_q\big[x^{\pm 1},y^{\pm 1}\big]$ of the quantum plane. After reproducing the partial results of a previous paper of the author related to symmetries with non-trivial action of the Cartan generator(s) of $U_q(\mathfrak{sl}_2)$ and the generic symmetries, a complete collection of non-generic symmetries is presented. Together, these collections constitute a complete list of $U_q(\mathfrak{sl}_2)$-symmetries on $\mathbb{C}_q\big[x^{\pm 1},y^{\pm 1}\big]$.