Self-distributive algebras and bialgebras

Abstract

We study 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 is given by rack and quandle bialgebras. But we go further to the general coassociative comultiplication. The principal motivation for this work is the development of linear algebra related to the notion of a quandle in analogy with the ubiquitous role of group algebras in the category of groups with possible applications to the theory of knot invariants. We describe self-distributive algebras and show that some quandle algebras and some Novikov algebras are self-distributive. We also give a full classification of counital self-distributive bialgebras in dimension \(2\) over \(\mathbb{C}\).