2-dimensional self-distributive non-counital bialgebras and knot invariants

TL;DR

This paper classifies all two-dimensional non-counital self-distributive bialgebras and explores their potential for constructing knot invariants, advancing the understanding of algebraic structures related to knot theory.

Contribution

It provides a complete classification of 2D non-counital self-distributive bialgebras and investigates their application in developing knot invariants.

Findings

All 2-dimensional non-counital self-distributive bialgebras are identified.

Analysis of these algebras for constructing quandles for knot invariants.

Part of a broader effort to linearize racks and quandles for representation theory.

Abstract

In the preprint of V. Bardakov, T. Kozlovskaya, D. Talalaev (Self-distributive bialgebras, arXiv:2501.19152) it was formulated a problem of classification of self-distributive bialgebras and was given classification of two-dimensional counital self-distributive bialgebras. In this paper, we consider non-counital case. We find all 2-dimensional algebras of this type. In constructed algebras we study the question of finding quandles for constructing knot invariants. This activity is part of the overall program for the linearization of the concepts of rack and quandle and the development of the representations theory of these structures.