arXiv subjects

Senne Trappeniers

Publications and source records attributed to Senne Trappeniers.

3 recordsLinked to original sources

arXiv2024

A Lazard correspondence for post-Lie rings and skew braces

We develop a Lazard correspondence between post-Lie rings and skew braces that satisfy a natural completeness condition. This is done through a thorough study of how the Lazard correspondence behaves on semi-direct sums of Lie rings. In particular, for a prime $p$ and $k<p$, we obtain a correspondence between skew braces of order $p^k$ and left nilpotent post-Lie rings of order $p^k$ on a nilpotent Lie ring. This therefore extends results by Smoktunowicz.

math.RA↗

arXiv2024

Indecomposable involutive set-theoretical solutions to the Yang-Baxter equation of size $p^2$

The quantum Yang-Baxter equation is a braiding condition on vector spaces which is of high relevance in several fields of mathematics, such as knot theory and quantum group theory. Their combinatorial counterpart are set-theoretic solutions to the Yang--Baxter equation, whose investigation is strongly driven by the study of algebraic objects called (skew) braces. In this article, we focus on indecomposable involutive non-degenerate set-theoretic solutions to the Yang-Baxter equation. More specifically, through a thorough analysis of their associated braces, we give a full classification of those which are of size $p^2$, for $p$ a prime.

math.QA↗

arXiv2023

Studying solutions of the Yang-Baxter equation through skew braces, with an application to indecomposable involutive solutions with abelian permutation group

We connect properties of set-theoretic solutions to the Yang--Baxter equation to properties of their permutation skew brace. In particular, a variation of the multipermutation level of a solution is presented and we show that it coincides with the multipermutation level of the permutation skew brace, contrary to the inequality that one has for the usual multipermutation level of solutions. We relate the number of orbits of a solution to generators of its permutation skew brace and relate different kinds of notions of generating sets of a skew brace. Also, the automorphism groups of solutions are studied through their permutation skew brace. As an application, we obtain a surprising result on subsolutions of multipermutation solutions and we give a description of all finite indecomposable involutive solutions to the Yang--Baxter equation with abelian permutation group. For multipermutation level 3, we obtain the precise number of isomorphism classes of such solutions of a given size.

math.RA↗