The Dynamical Yang-Baxter Relation and the Minimal Representation of the Elliptic Quantum Group

TL;DR

This paper derives explicit minimal L-matrices linked to elliptic quantum groups and IRF models, revealing their algebraic structures and solutions to the Yang-Baxter equation, advancing understanding of integrable systems.

Contribution

It provides explicit forms of minimal L-matrices for elliptic quantum groups and analyzes their algebraic properties, including solutions to the Yang-Baxter equation.

Findings

Explicit L-matrix forms with spectral parameter dependence.

Algebra of form A is trivial; algebra of form B satisfies YBE.

PBW basis and centers for algebra of form B established.

Abstract

In this paper, we give the general forms of the minimal $L$ matrix (the elements of the $L$-matrix are $c$ numbers) associated with the Boltzmann weights of the $A_{n-1}^1$ interaction-round-a-face (IRF) model and the minimal representation of the $A_{n-1}$ series elliptic quantum group given by Felder and Varchenko. The explicit dependence of elements of $L$-matrices on spectral parameter $z$ are given. They are of five different forms (A(1-4) and B). The algebra for the coefficients (which do not depend on $z$) are given. The algebra of form A is proved to be trivial, while that of form B obey Yang-Baxter equation (YBE). We also give the PBW base and the centers for the algebra of form B.