Colored Vertex Models, Colored IRF Models and Invariants of Trivalent Colored Graphs

TL;DR

This paper develops formulas for quantum group coefficients at roots of unity, explores colored vertex and IRF models, and constructs invariants of trivalent colored graphs using $U_q(sl(2))$ representations.

Contribution

It introduces explicit formulas for Clebsch-Gordan and Racah coefficients at roots of unity and constructs new invariants of colored graphs from quantum group representations.

Findings

Formulas for Clebsch-Gordan coefficients at roots of unity

Formulas for Racah coefficients at roots of unity

Construction of invariants of trivalent colored graphs

Abstract

We present formulas for the Clebsch-Gordan coefficients and the Racah coefficients for the root of unity representations ($N$-dimensional representations with $q^{2N}=1$) of $U_q(sl(2))$. We discuss colored vertex models and colored IRF (Interaction Round a Face) models from the color representations of $U_q(sl(2))$. We construct invariants of trivalent colored oriented framed graphs from color representations of $U_q(sl(2))$.