More on $U_q(su(1,1))$ with $q$ a Root of Unity

TL;DR

This paper explores the structure and representations of the quantum algebra $U_q(su(1,1))$ at roots of unity, including decomposition formulas, connections to $SL(2,R)$, supersymmetry, and explicit realizations of superalgebras.

Contribution

It provides a detailed analysis of highest weight modules, Clebsch-Gordan decompositions, and new realizations of supersymmetry and superalgebras at roots of unity.

Findings

Detailed structure of irreducible highest weight modules

Clebsch-Gordan decomposition for tensor products

Explicit realization of $Osp(1|2)$ at $q^2=-1$

Abstract

Highest weight representations of $U_q(su(1,1))$ with $q=\exp \pi i/N$ are investigated. The structures of the irreducible hieghesat weight modules are discussed in detail. The Clebsch-Gordan decomposition for the tensor product of two irreducible representations is discussed. By using the results, a representation of $SL(2,R)\otimes U_q(su(2))$ is also presented in terms of holomorphic sections which also have $U_q(su(2))$ index. Furthermore we realise $Z_N$-graded supersymmetry in terms of the representation. An explicit realization of $Osp(1 \vert 2)$ via the heighest weight representation of $U_q(su(1,1))$ with $q^2=-1$ is given.