Searcharxiv⌕ Search

arXiv subjects

Carsten Dietzel

Publications and source records attributed to Carsten Dietzel.

15 recordsLinked to original sources
arXiv2026

Proof of Rump's Retraction Conjecture for Quasilinear Cycle Sets

Nondegenerate cycle sets were introduced by Rump as an algebraic framework for nondegenerate, involutive solutions to the Yang--Baxter equation. Nondegenerate cycle set structures on abelian groups, such as translation-invariant and quasilinear cycle sets, are of particular interest when studying the retraction problem in the theory of the Yang--Baxter equation. In this article, we solve the retraction problem for finite quasilinear cycle sets by showing that each nontrivial quasilinear cycle set is retractable, thus proving a conjecture of Rump.

math.QA↗
arXiv2026

Non-degeneracy of Killing forms on real conjugacy classes of finite groups

Killing forms on finite groups arise as special cases of braided Killing forms on braided Lie algebras. If $\mathcal{C}$ is a conjugation-stable subset of a finite group $G$, the Killing form on $\mathbb{C}\mathcal{C}$ is given by $K_\mathcal{C}(a,b) = |C_G(ab) \cap \mathcal{C}|$ for $a,b \in \mathcal{C}$. It is conjectured in previous work by L\'opez Pe\~na, Majid and Rietsch that $K_\mathcal{C}$ is non-degenerate for any real conjugacy class $\mathcal{C}$ in a finite simple group. In this article, we reformulate the conjecture and introduce combinatorial conditions - the $\textit{1-element condition}$ and the $\textit{2-element condition}$ - that are sufficient for non-degeneracy to hold. This allows us to prove the conjecture for simple groups of the form $\mathrm{PSL}_2(q)$ and certain conjugacy classes in the alternating and symmetric groups. Moreover, we verify computationally that every real conjugacy class in a simple group of order $\leq 10^9$ fulfills at least one of these two conditions, thereby significantly extending the computational evidence for the conjecture. This raises the question whether these conditions are satisfied by all conjugacy classes in finite simple groups.

math.GR↗
arXiv2025

How structure groups and monoids grow

The structure groups and monoids of set-theoretic solutions to the Yang-Baxter Equation can be regarded as deformations of free abelian groups resp. monoids. In this work, we obtain explicit formulae for the growth series of the structure groups and monoids of transposition and dihedral quandles, and of the structure groups of permutation quandles. These quandles provide important families of YBE solutions. The intricate nature of our formulae confirms that, while preserving many nice properties of free abelian groups, even the simplest structure groups and monoids are remarkably rich objects. We also establish some structural properties and easily computable normal forms for the monoids considered.

math.GR↗
arXiv2025

Groups of $\mathrm{I}_G$-type

In this work, we address a question posed by Dehornoy et al. in the book "Foundations of Garside Theory" that asks for a theory of groups of $\mathrm{I}_G$-type when $G$ is a Garside group. In this article, we introduce a broader notion than the one suggested by Dehornoy et al.: given a left-ordered group $G$, we define a group of $\mathrm{I}_G$-type as a left-ordered group whose partial order is isomorphic to those of $G$. Furthermore, we develop methods to give a characterization of groups of $\mathrm{I}_{\Gamma}$-type in terms of skew braces when $\Gamma$ is an Artin-Tits group of spherical type and classify all groups of $\mathrm{I}_{\Gamma}$-type where $\Gamma$ is an irreducible spherical Artin-Tits group, therefore providing an answer to another question of Dehornoy et al. concerning $\mathrm{I}_{B_n}$ structures where $B_n$ is the braid group on $n$ strands with its canonical Garside structure.

math.GR↗
arXiv2025

Endocabling of involutive solutions to the Yang-Baxter equation, with an application to solutions whose diagonal is a cyclic permutation

In this article, we introduce endocabling as a technique to deform involutive, non-degenerate set-theoretic solutions to the Yang-Baxter equation (``solutions'', for short) by means of $\lambda$-endomorphisms of their associated permutation brace, thus generalizing the cabling method by Lebed, Vendramin and Ram\'{i}rez. In the first part of the article, we define endocabling and investigate the behaviour of solutions and their invariants under endocabling. In the second part, we apply our findings to solutions of size $n$ whose diagonal map is an $n$-cycle: we will prove that solutions with this property whose size is an odd prime power, are of finite multipermutation level. Furthermore, solutions with this property whose size is a power of $2$, will be proven either to be of finite multipermutation level or to admit an iterated retraction onto a unique solution of size $4$. We formulate our results in the language of cycle sets.

math.QA↗
arXiv2024

Coprime extensions of indecomposable solutions to the Yang-Baxter equation

In this article, we introduce a method to extend involutive nondegenerate set-theoretic solutions to the Yang--Baxter equation by means of equivariant mappings to graded modules, thus leading to the notion of a twisted extension. Furthermore, we define coprime extensions of solutions and prove that each coprime extension of indecomposable solutions can be obtained as a suitable twisted extension. We then apply our results to obtain a full description of indecomposable solutions of size $pqr$, where $p,q,r$ are different primes, from a structure theorem of Ced\'o and Okni\'nski. We close with some remarks on a cohomology theory for solutions developed by Lebed and Vendramin. We express our results in the language of cycle sets.

math.QA↗
arXiv2024

On Dehornoy's representation for the Yang-Baxter equation

This article investigates Dehornoy's monomial representations for structure groups and Coxeter-like groups associated with a set-theoretic solution to the Yang--Baxter equation. Using the brace structure of these groups and the language of cycle sets, we prove that the irreducibility of the associated monomial representations is equivalent to the indecomposability of the underlying solutions, except when the Dehornoy class is two. For indecomposable solutions, we show that these representations are induced from certain explicitly constructed one-dimensional representations.

math.GR↗
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

Decomposition and Structure theorems for Garside-like groups with modular lattice structure

Despite being a vast generalization of Garside groups, right $\ell$-groups with noetherian lattice structure and strong order unit share a lot of the properties of Garside groups. In the present work, we prove that every modular noetherian right $\ell$-group with strong order unit decomposes as a direct product of beams, which are sublattices that correspond to the directly indecomposable factors of the strong order interval. Furthermore, we show that the beams of dimension $δ\geq 4$ can be coordinatized by the $R$-lattices in $Q^δ$, where $Q$ is a noncommutative discrete valuation field with valuation ring $R$. In particular, this gives a precise description of a very big family of modular Garside groups.

math.GR↗
arXiv2020

Right $\ell$-groups associated with von Neumann algebras

In [Rum18], Rump defined and characterized noncommutative universal groups $G(X)$ for generalized orthomodular lattices $X$. We give an explicit construction of $G(X)$ in terms of \emph{pure paraunitary groups} when $X$ is the projection lattice of a von Neumann algebra. The results given here extend some of those in [Die19].

math.QA↗
arXiv2018

On a theorem of Hildebrand

We prove that for each multiplicative subgroup $A$ of finite index in $\mathbb{Q}^+$, the set of integers $a$ with $a, a+1 \in A$ is an IP-set. This generalizes a theorem of Hildebrand concerning completely multiplicative functions taking values in the $k$-th roots of unity.

math.NT↗
arXiv2018

A right-invariant lattice-order on groups of paraunitary matrices

In \cite{rump_goml}, Rump defined and characterized noncommutative universal groups $G(X)$ for generalized orthomodular lattices $X$. We give an explicit description of $G(X)$ in terms of \emph{paraunitary} matrix groups, whenever $X$ is the orthomodular lattice of subspaces of a finite-dimensional $k$-vector space $V$ that is equipped with an anisotropic, symmetric $k$-bilinear form.

math.QA↗
arXiv2018

Braces of order $p^2q$

In this article we classify the left braces of order $p^2q$ where $p,q$ are primes fulfilling $q > p+1$. This classification includes a proof of three conjectures of Guarnieri and Vendramin (\cite[Conjectures 6.2-6.4]{Vendramin_skew}) concerning the number of isomorphism classes of left braces of order $p^2q$ for certain values of $p,q$.

math.QA↗