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TMF, 2025 Volume 224, Number 1, Pages 3–21 (Mi tmf10900)

Self-distributive algebras and bialgebras

V. G. Bardakovabc, T. A. Kozlovskayac, D. V. Talalaevde

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State Agrarian University, Novosibirsk, Russia
c Regional Scientific and Educational Mathematical Center of Tomsk State University, Tomsk, Russia
d Lomonosov Moscow State University, Moscow, Russia
e Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, Yaroslavl, Russia

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 $\mathbb{C}$.

Keywords: algebra, coalgebra, bialgebra, rack, quandle, rack algebra, rack bialgebra, self-distributivity, Yang–Baxter equation.

MSC: 17A30

Received: 29.01.2025
Revised: 01.04.2025

DOI: 10.4213/tmf10900



© Steklov Math. Inst. of RAS, 2025