Nonstandard q-deformation of the universal enveloping algebra $U'({\rm so}_n)$

TL;DR

This paper introduces a nonstandard q-deformation of the universal enveloping algebra of so(n), explores its properties, and classifies its irreducible representations at roots of unity, revealing new algebraic structures.

Contribution

It defines and analyzes a nonstandard q-deformation of U(so(n)), distinct from the standard quantum algebra, and classifies its irreducible representations at roots of unity.

Findings

Irreducible representations act on p^N-dimensional spaces.

Representations depend on r=dim so(n) complex parameters.

The algebra differs from the Drinfeld--Jimbo quantum algebra.

Abstract

We describe properties of the nonstandard q-deformation $U'_q({\rm so}_n)$ of the universal enveloping algebra $U({\rm so}_n)$ of the Lie algebra ${\rm so}_n$ which does not coincide with the Drinfeld--Jimbo quantum algebra $U_q({\rm so}_n)$. Irreducible representations of this algebras for q a root of unity q^p=1 are given. These representations act on p^N-dimensional linear space (where N is a number of positive roots of the Lie algebra ${\rm so}_n$) and are given by $r={\rm dim} {\rm so}_n$ complex parameters.