$U_q\left(\mathfrak{sl}_2\right)$-symmetries of the quantum disc: a complete list

TL;DR

This paper classifies all $U_q(sl_2)$-symmetries on the quantum disc, introducing the grading jump as a key invariant, which can only take three values, and identifies compatible symmetries with involutions.

Contribution

It provides a complete classification of $U_q(sl_2)$-symmetries on the quantum disc, introducing the grading jump invariant and analyzing symmetry compatibility conditions.

Findings

Grading jump can only be 0, 1, or -1.

Complete list of symmetries classified based on grading jump.

Identified symmetries satisfying involution compatibility.

Abstract

This work presents a classification of $U_q(\mathfrak{sl}_2)$-symmetries on the quantum disc. The principal invariant of such classification, the grading jump, is introduced. It turns out that, under the present subjects, the grading jump can take only 3 values: $0$, $1$, $-1$. The subcollection of the complete collection of symmetries is extracted in such a way that the selected symmetries satisfy certain compatibility condition for involutions.