Generic symmetries of the Laurent extension of quantum plane

TL;DR

This paper classifies various algebraic structures of the quantum plane extended by Laurent polynomials under quantum group actions, revealing uncountably many distinct module algebra structures.

Contribution

It provides a comprehensive list of $U_q(sl_2)$-module algebra structures on the Laurent extension of the quantum plane, including those with non-constant Cartan actions.

Findings

Uncountably many isomorphism classes of module algebra structures.

Complete classification including non-constant Cartan actions.

Explicit descriptions of algebra structures under quantum group actions.

Abstract

A list of generic $U_q(sl_2)$-module algebra structures on the Laurent polynomial algebra over the quantum plane with uncountably many isomorphism classes is produced. Also, a complete list of such structures is presented in which the action of Cartan generator of $U_q(sl_2)$ does not reduce to multiplying x and y (the generators of quantum plane) by constants.