The method of vacuum vectors in the theory of Yang - Baxter equation

TL;DR

This paper introduces a novel approach using vacuum vectors of L-operators to construct solutions to the Yang-Baxter equation without a traditional spectral parameter, linking solutions at roots of unity to algebraic curves and the chiral Potts model.

Contribution

It is the first to analyze solutions of the Yang-Baxter equation at roots of unity and relate them to algebraic curves of genus greater than one, expanding the understanding of integrable models.

Findings

Solutions related to algebraic curves of genus >1

Connection to the chiral Potts model

Construction of spectral-parameter-free solutions

Abstract

In modern terminology, this is the first published paper where the solutions of Yang - Baxter equation "at roots of unity" were analyzed and shown to be related to algebraic curves of genus >1. They are also known now to be connected with the "chiral Potts model". The paper's abstract as written in 1986 reads: "Vacuum vectors of an L-operator form a holomorphic bundle over the vacuum curve of that operator. These notions, as well as the theory of commutation relations of the 6-vertex model, are used in this work for constructing solutions of the Yang - Baxter equation that do not possess a spectral parameter of traditional type".