Yang-Baxter operators in symmetric categories

TL;DR

This paper extends the theory of Yang-Baxter operators to symmetric monoidal categories, introducing new classes of solutions, and generalizes existing algebraic structures like braces and matched pairs within this broader framework.

Contribution

It develops a categorical framework for Yang-Baxter solutions, including infinite non-set-theoretical examples, and unifies structures like braces and cocycle-based solutions.

Findings

Classifies non-degenerate solutions via invertible 1-cocycles.

Identifies infinite families of solutions beyond set-theoretical ones.

Generalizes braces and matched pairs to symmetric monoidal categories.

Abstract

We introduce non-degenerate solutions of the Yang-Baxter equation in the setting of symmetric monoidal categories. Our theory includes non-degenerate set-theoretical solutions as basic examples. However, infinite families of non-degenerate solutions (that are not of set-theoretical type) appear. As in the classical theory of Etingof, Schedler and Soloviev, non-degenerate solutions are classified in terms of invertible 1-cocycles. Braces and matched pairs of cocommutative Hopf algebras (or braiding operators) are also generalized to the context of symmetric monoidal categories and turn out to be equivalent to invertible 1-cocycles.