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

TL;DR

The paper delivers a complete classification of $2$-dimensional non-counital self-distributive bialgebras by first classifying all possible non-counital coassociative coproducts on a $2$-D space, then determining compatible multiplications that satisfy both the algebra and self-distributivity axioms. It also analyzes idempotents and quandles inside these algebras, illustrating quandle-like substructures and highlighting limitations on rack structures in this small dimension. The results extend the counital $2$-D classification and advance the program of linearizing racks and quandles, with implications for knot invariants derived from quandle rings and their idempotents. The work provides explicit families of multiplications (e.g., various $M1$–$M4$ cases) corresponding to distinct coproduct types, and clarifies when quandle-like structures arise or fail in these $2$-D settings.

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.

Theorems & Definitions (16)

• Definition 2.1
• Proposition 2.2
• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Proposition 3.6
• Proposition 3.7