Irrational Uq(sl2)-symmetries on the quantum plane

TL;DR

This paper classifies all Uq(sl2)-symmetries of the quantum plane using symbolic matrices, revealing uncountably many classes and describing their composition series and classical limits.

Contribution

It provides a complete classification of Uq(sl2)-symmetries on the quantum plane, introducing a novel symbolic matrix method for labeling actions.

Findings

Uncountably many isomorphism classes of symmetries identified

Complete description of composition series of representations

Classical limits of the symmetries are characterized

Abstract

The main result of this work is to present the complete list of Uq(sl2)-symmetries of quantum plane. For that, the structure of quantum plane automorphisms is used. Our idea in classifying the above symmetries is in introducing some special symbolic matrices to label the series of actions in question. The matrices depict which generators of Uq(sl2) produce nonzero images by acting on the generators of quantum plane. This data determines the weights of an action unambiguously. It turns out that there are uncountably many isomorphism classes of the symmetries. Another result is in describing composition series of the corresponding representations. The classical limit of the above symmetries is given.