Self-distributive algebras and bialgebras

TL;DR

This paper explores self-distributive algebraic structures, including algebras and bialgebras, their relations to other algebraic systems, and provides classifications and examples relevant to knot invariants and algebraic theory.

Contribution

It introduces a comprehensive study of self-distributive algebras and bialgebras, including classifications and connections to other algebraic structures.

Findings

Some quandle algebras are self-distributive

Some Novikov algebras are self-distributive

Full classification of 2-dimensional counital self-distributive bialgebras over C

Abstract

This article is devoted to the study of self-distributive algebraic structures: algebras, bialgebras; additional structures on them, relations of these structures with Hopf algebras, Lie algebras, Leibnitz algebras etc. The basic example of such structures are rack- and quandle bialgebras. But we go further - to the general coassociative comultiplication. The principal motivation for this work is the development of the linear algebra related with a notion of a quandle in analogy with the ubiquitous role of group algebras in the category of groups with perspective applications to the theory of knot invariants. We give description of self-distributive algebras and show that some quandle algebras and some Novikov algebras are self-distributive. Also, we give a full classification of counital self-distributive bialgebras in dimension 2 over C.