Fusion Rules of the Lowest Weight Representations of osp_q(1|2) at Roots of Unity: Polynomial Realization and Degeneration at Roots of Unity

TL;DR

This paper investigates the structure and fusion rules of lowest weight representations of the quantum superalgebra osp_q(1|2) at roots of unity, using polynomial realizations and analyzing indecomposable representations.

Contribution

It provides a complete description of fusion rules and the emergence of indecomposable representations at roots of unity for osp_q(1|2).

Findings

Complete fusion rules at roots of unity are derived.

Realization of generators as finite-difference operators is demonstrated.

Indecomposable representations appear in tensor product decompositions.

Abstract

The degeneracy of the lowest weight representations of the quantum superalgebra $osp_q(1|2)$ and their tensor products at exceptional values of %when deformation parameter $q$ takes exceptional values is studied. The main features of the structures of the finite dimensional lowest weight representations and their fusion rules are illustrated using realization of group generators as finite-difference operators acting in the space of the polynomials. The complete fusion rules for the decompositions of the tensor products at roots of unity are presented. The appearance of indecomposable representations in the fusions is described using Clebsh-Gordan coefficients derived for general values of $q$ and at roots of unity.