q-deformed algebras $U_q(so_n)$ and their representations

TL;DR

This paper constructs explicit finite-dimensional irreducible representations of nonstandard q-deformed algebras U_q(so_n) using a q-analogue of the Gel'fand-Tsetlin basis, confirming their algebraic relations for generic q.

Contribution

It provides a detailed construction and proof of finite-dimensional irreducible representations of U_q(so_n) in a Gel'fand-Tsetlin basis, extending classical representation theory to nonstandard q-deformations.

Findings

Explicit finite-dimensional irreducible representations constructed

Representation operators satisfy trilinear relations for generic q

Gel'fand-Tsetlin basis effectively extends to q-deformed algebras

Abstract

For the nonstandard $q$-deformed algebras $U_q(so_n)$, defined recently in terms of trilinear relations for generating elements, most general finite dimensional irreducible representations directly corresponding to those of nondeformed algebras $so(n)$ (i.e., characterized by the same sets of only integers or only half-integers as in highest weights of the latter) are given explicitly in a $q$-analogue of Gel'fand-Tsetlin basis. Detailed proof, for $q$ not equal to a root of unity, that representation operators indeed satisfy relevant (trilinear) relations and define finite dimensional irreducible representations is presented. The results show perfect suitability of the Gel'fand-Tsetlin formalism concerning (nonstandard) $q$-deformation of $so(n)$.