Representations of $SO(5)_{q}$ and Non-Minimal $q$-Deformation

TL;DR

This paper explores different bases for representations of the quantum group $SO(5)_q$, revealing that non-minimal deformations are necessary in some cases and discussing their implications.

Contribution

It introduces the concept of non-minimal $q$-deformations in $SO(5)_q$ representations and compares different basis constructions.

Findings

Non-minimal deformation is essential in certain $SO(5)_q$ representations.

Different bases lead to distinct $q$-deformation behaviors.

Parallel use of bases offers broader insights into quantum group representations.

Abstract

Representations of $SO(5)_{q}$ can be constructed on bases such that either the Chevalley triplet $(e_{1},\;f_{1},\;h_{1})$ or $(e_{2},\;f_{2},\;h_{2})$ has the standard $SU(2)_{q}$ matrix elements. The other triplet in each cases has a more complicated action. The $q$-deformation of such representations present striking differences. In one case a {\bf non-minimal} deformation is found to be essential. This is explained and illustrated below. Broader interests of a parallel use of the two bases are pointed out.