Self-distributive algebras and bialgebras
Valeriy G. Bardakov, Tatiana A. Kozlovskaya, Dmitry V. Talalaev
TL;DR
The paper investigates self-distributive structures in algebras and coalgebras, focusing on rack and quandle-based constructions and their bialgebra generalizations. It develops linear rack/quandle frameworks and establishes Yang–Baxter solutions arising from linear racks, including adjoint-action constructions on cocommutative Hopf algebras. A key result is the full classification of 2-dimensional counital self-distributive bialgebras over fields of characteristic not equal to $2$, organized by three comultiplication types and explicit multiplication tables. The work elucidates connections to generalized Jordan algebras and knot-invariant related algebraic structures, providing a concrete low-dimensional landscape for self-distributive algebraic systems with potential applications in representation theory and low-dimensional topology.
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.