Cyclic monodromy matrices for sl(n) trigonometric R-matrices

TL;DR

This paper investigates the algebraic structure of monodromy matrices associated with sl(n) trigonometric R-matrices at roots of unity, introducing factorizable representations and their intertwiners linked to the sl(n) chiral Potts model.

Contribution

It introduces a new class of factorizable representations for the algebra of monodromy matrices and expresses intertwiners through Boltzmann weights, advancing understanding of these algebraic structures.

Findings

Finite-dimensional cyclic irreducible polynomial representations are characterized.

Intertwiners for cocommuting factorizable representations are constructed.

Connections to the sl(n) chiral Potts model are established.

Abstract

The algebra of monodromy matrices for sl(n) trigonometric R-matrices is studied. It is shown that a generic finite-dimensional polynomial irreducible representation of this algebra is equivalent to a tensor product of L-operators. Cocommutativity of representations is discussed. A special class of representations - factorizable representations is introduced and intertwiners for cocommuting factorizable representations are written through the Boltzmann weights of the sl(n) chiral Potts model. Let us consider an algebra generated by noncommutative entries of the matrix $T(u)$ satisfying the famous bilinear relation originated from the quantum inverse scattering method: $R(\la-\mu)T(\la)T(\mu)=T(\mu)T(\la) R(\la-\mu)$ where $R(\la)$ is R-matrix. For historical reasons this algebra is called the algebra of monodromy matrices. If $\g$ is a simple finite-dimensional Lie algebra and $R(\la)$ is $\g$-invariant R-matrix the algebra of monodromy matrices after a proper specialization gives the Yangian $Y(\g)$ introduced by Drinfeld. If $R(\la)$ is corresponding trigonometric R-matrix this algebra is closely connected with $U_q(\g)$ and $U_q(\hat\g)$ at zero level. If $R(\la)$ is $sl(2)$ elliptic R-matrix the algebra of monodromy matrices gives rise to Sklyanin's algebra. In this paper we shall study algebras of monodromy matrices for $sl(n)$ trigonometric R-matrices at roots of 1. Finite-dimensional cyclic irreducible polynomial representations and their intertwiners are discussed.