New fusion rules and $\cR$-matrices for $SL(N)_q$ at roots of unity

TL;DR

This paper develops new fusion rules and $ $-matrices for $SL(N)_q$ at roots of unity, enabling a better understanding of tensor product decompositions and intertwiners in quantum group representations.

Contribution

It introduces novel fusion rules and heterogeneous $ $-matrices for $SL(N)_q$ at roots of unity, expanding the tools for analyzing quantum group representations.

Findings

Derived full reducibility into semi-periodic representations.

Constructed $ $-matrices satisfying all Yang-Baxter equations.

Established compatibility between different types of representations.

Abstract

We derive fusion rules for the composition of $q$-deformed classical representations (arising in tensor products of the fundamental representation) with semi-periodic representations of $SL(N)_q$ at roots of unity. We obtain full reducibility into semi-periodic representations. On the other hand, heterogeneous $\cR$-matrices which intertwine tensor products of periodic or semi-periodic representations with $q$-deformed classical representations are given. These $\cR$-matrices satisfy all the possible Yang Baxter equations with one another and, when they exist, with the $\cR$-matrices intertwining homogeneous tensor products of periodic or semi-periodic representations. This compatibility between these two kinds of representations has never been used in physical models.