$SO(5)_{q}$ and Contraction

TL;DR

This paper constructs explicit representations of the quantum group $SO(5)_q$ for all q, including roots of unity, and provides a contraction method to obtain non-semisimple Hopf algebras with explicit Casimir operators.

Contribution

It introduces a comprehensive construction of $SO(5)_q$ representations on the Chevalley basis and a contraction procedure for non-semisimple cases, unifying generic and root of unity cases.

Findings

Explicit matrix elements for all representations up to dimension 4.

A contraction method yielding complete Hopf algebras for non-semisimple cases.

Explicit evaluation of the $q$-deformed quadratic Casimir operator.

Abstract

Representations of $SO(5)_{q}$ are constructed explicitly on the Chevalley basis for all $q$, generic and root of unity. Matrix elements of the generators are obtained for all representations depending on three variable indices, the maximal number being 4. A prescription for contraction is given such that a complete Hopf algebra is immediately obtained for the non-semisimple contracted case. For $q$ a root of unity the periodic representations for $SO(5)_{q}$ and the contracted algebra are obtained directly in the "fractional part" formalism which unifies the treatments for the generic and root of unity cases. The $q$-deformed quadratic Casimir operator is explicitly evaluated for the representations presented.