A universal-algebra and combinatorial approach to the set-theoretic Yang-Baxter equation

TL;DR

This paper introduces a new algebraic framework using universal algebra and combinatorics to classify and analyze solutions to the set-theoretic Yang-Baxter equation, revealing countably many simple solutions linked to ergodic theory.

Contribution

It develops the concept of P{ }lonka bi-magmas to classify BLS-type solutions and establishes categorical adjunctions among solution classes.

Findings

Countably many isomorphism classes of simple BLS-solutions.

All simple solutions are describable via odometer transformations.

New classes of YB-solutions, such as bi-connected and simple, are studied and classified.

Abstract

We introduce a new variety of set-theoretic non-associative algebras, P{\l}onka bi-magmas, to describe and classify all solutions of the set-theoretic Yang-Baxter (YB) equation of Baaj-Long-Skandalis (BLS) type. We also study new classes of YB-solutions (bi-connected, simple), and classify the BLS-solutions that fit into those classes. There are only countably many isomorphism classes of simple BLS-solutions, for instance, and they are all describable in terms of the odometer transformations familiar from ergodic theory. Placing Drinfel'd's problem of classifying the set-theoretic solutions of the Yang-Baxter equation in a universal-algebra context with a combinatorial flavor, we also prove the existence of adjunctions between various categories of solutions.