Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2502.16875

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

Valeriy G. Bardakov·Tatiana A. Kozlovskaya·Alexander S. Panasenko·Dmitry V. Talalaev
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.QAmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Valeriy G. Bardakov, Tatiana A. Kozlovskaya, Alexander S. Panasenko, Dmitry V. Talalaev. 2025-02-24. 2-dimensional self-distributive non-counital bialgebras and knot invariants. https://arxiv.org/abs/2502.16875

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Categorification of quasi-split iquantum groups

We introduce a new family of graded 2-categories generalizing the 2-quantum groups introduced by Khovanov, Lauda and Rouquier. We use them to categorify quasi-split iquantum groups in all symmetric types.

math.QA↗
arXiv2026

The Ring of Differential Operators on a Nodal Curve is not a Bialgebroid

In a previous article, we showed that local projectivity is a sufficient condition for the existence of a bialgebroid structure on the ring of differential operators on an affine variety. In this note, we show using elementary methods that the ring of differential operators on a nodal curve is neither locally projective nor does it admit a bialgebroid structure.

math.QA↗
arXiv2026

Coset representatives corresponding to Yetter-Drinfeld modules of modular group and continued fraction

We give complete conjugacy classes of modular group SL(2,Z). Particularly, the conjugacy classes of hyperbolic elements are decided by the proper equivalence classes of indefinite forms, and we give an example. Finally, we describe the coset representatives of centralizer of S, ST, T and hyperbolic elements of SL(2,Z) by regular continued fraction. In conclusion, most Nichols algebras over modular group are infinite-dimensional except Proposition 4.10.

math.QA↗