Self-distributive algebras and bialgebras
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.
Cite
@article{arxiv.2501.19152,
title = {Self-distributive algebras and bialgebras},
author = {Valeriy G. Bardakov and Tatiana A. Kozlovskaya and Dmitry V. Talalaev},
journal= {arXiv preprint arXiv:2501.19152},
year = {2025}
}