Distributivity versus associativity in the homology theory of algebraic structures

TL;DR

This paper explores the homology theory of non-associative distributive algebraic structures, providing a general framework, computing examples, and discussing potential links to Khovanov homology.

Contribution

It develops a comprehensive framework for homology of distributive structures, extending previous work and including explicit computations and potential applications.

Findings

Computed examples of 1-term, 2-term, and 3-term homology.

Discussed 4-term homology for Boolean algebras and lattices.

Outlined possible connections to Khovanov homology.

Abstract

While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, were neglected until recently. Distributive structures have been studied for a long time. In 1880, C.S. Peirce emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only sixteen years ago by Fenn, Rourke, and Sanderson. We develop this theory in the historical context and propose a general framework to study homology of distributive structures. We illustrate the theory by computing some examples of 1-term, 2-term, and 3-term homology, and then discussing 4-term homology for Boolean algebras and distributive lattices. We outline potential relations to Khovanov homology, via the Yang-Baxter operator.