Polynomial Realization of $s\ell_q(2)$ and Fusion Rules at Exceptional Values of $q$

TL;DR

This paper constructs polynomial representations of the quantum algebra s ext{l}_q(2) at roots of unity and explores fusion rules, providing explicit examples and conjectures for the general case.

Contribution

It introduces polynomial realizations of s ext{l}_q(2) at roots of unity and formulates a conjecture for the general fusion rules based on Casimir eigenvalues.

Findings

Explicit polynomial representations at q^N=1

Fusion rules illustrated with simple cases

Conjecture for general fusion rules

Abstract

Representations of the $s\ell_q(2)$ algebra are constructed in the space of polynomials of real (complex) variable for $q^N=1$. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and conjecture for general case is formulated.