Cohomology of Categorical Self-Distributivity

TL;DR

This paper develops a cohomology theory for self-distributive structures in coalgebra categories, linking them to solutions of the Yang--Baxter equation and unifying Lie algebra and quandle cohomologies.

Contribution

It introduces a new cohomology framework for self-distributive structures in coalgebra categories, connecting them to Yang--Baxter solutions and existing cohomologies.

Findings

Constructed examples from vector spaces, Lie algebras, and Hopf algebras.

Provided a cohomology theory unifying Lie algebra and quandle cohomologies.

Demonstrated applications to deformations of self-distributive structures.

Abstract

We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive operations of these structures provide solutions of the Yang--Baxter equation, and, conversely, solutions of the Yang--Baxter equation can be used to construct self-distributive operations in certain categories. Moreover, we present a cohomology theory that encompasses both Lie algebra and quandle cohomologies, is analogous to Hochschild cohomology, and can be used to study deformations of these self-distributive structures. All of the work here is informed via diagrammatic computations.