Central elements of the elliptic $Z_n$ monodromy matrix algebra at roots of unity

TL;DR

This paper investigates the structure of the center of the algebra of monodromy matrices associated with the $ Z_n$ R-matrix at special rational crossing parameters, revealing an enlarged center at roots of unity, generalizing previous results.

Contribution

It extends the understanding of the central elements of elliptic $Z_n$ monodromy matrix algebra at roots of unity, generalizing known results to higher rank cases.

Findings

Center is larger at rational crossing parameters $w=n/N$

Generalizes Belavin and Jimbo's results to higher rank

Connects elliptic algebra structure with quantum groups at roots of unity

Abstract

The central elements of the algebra of monodromy matrices associated with the $\mathbb{Z}_n$ R-matrix are studied. When the crossing parameter $w$ takes a special rational value $w=\frac{n}{N}$, where $N$ and $n$ are positive coprime integers, the center is substantially larger than that in the generic case for which the "quantum determinant" provides the center. In the trigonometric limit, the situation corresponds to the quantum group at roots of unity. This is a higher rank generalization of the recent results by Belavin and Jimbo.