arXiv subjects

Magdalena Wiertel

Publications and source records attributed to Magdalena Wiertel.

6 recordsLinked to original sources

arXiv2026

Indecomposable solutions with permutation brace of size $p^3$ and permutation braces of small sizes

Using the construction of Bachiller, Ced\'o and Jespers, we give a complete classification of the indecomposable involutive set-theoretic solutions to the Yang-Baxter equation whose permutation brace has size $p^3$, where p is an odd prime. We also give an algorithm for systematically producing all indecomposable involutive solutions with a given permutation brace, and use this to enumerate these with the permutation brace of size up to 107.

math.GR↗

arXiv2025

Quasi racks, quasi bijective and quasi non-degenerate set-theoretic solutions of the Yang-Baxter equation

This work initiates a systematic study of the class of quasi bijective and quasi non-degenerate solutions to the set-theoretic Yang-Baxter equation. The motivation stems from the observation that solutions that arise from dual weak braces belong to these classes. The notions of quasi rack and derived solution are introduced and examined, extending the classical definitions. Additionally, a family of quasi left non-degenerate solutions is described in terms of quasi racks and g-twists, analogous to the left non-degenerate case. Furthermore, we completely characterize a class of quasi racks that are Plonka sum of racks.

math.QA↗

arXiv2023

The Gelfand-Kirillov dimension of Hecke-Kiselman algebras

Hecke-Kiselman algebras $A_Θ$, over a field $k$, associated to finite oriented graphs $Θ$ are considered. It has been known that every such algebra is an automaton algebra in the sense of Ufranovskii. In particular, its Gelfand-Kirillov dimension is an integer if it is finite. In this paper, a numerical invariant of the graph $Θ$ that determines the dimension of $A_Θ$ is found. Namely, we prove that the Gelfand-Kirillov dimension of $A_Θ$ is the sum of the number of cyclic subgraphs of $Θ$ and the number of oriented paths of a special type in the graph, each counted certain specific number of times.

math.RA↗

arXiv2021

Irreducible representations of Hecke-Kiselman monoids

Let $K[HK_Θ]$ denote the Hecke-Kiselman algebra of a finite oriented graph $Θ$ over an algebraically closed field $K$. All irreducible representations, and the corresponding maximal ideals of $K[HK_Θ]$, are characterized in case this algebra satisfies a polynomial identity. The latter condition corresponds to a simple condition that can be expressed in terms of the graph $Θ$. The result shows a surprising similarity to the classical results on representations of finite semigroups; namely every representation either comes form an idempotent in the Hecke-Kiselman monoid $HK_Θ$ (and hence it is $1$-dimensional), or it comes from certain semigroup of matrix type (which is an order in a completely $0$-simple semigroup over an infinite cyclic group). The case when $Θ$ is an oriented cycle plays a crucial role; the prime spectrum of $K[HK_Θ]$ is completely characterized in this case.

math.RT↗

arXiv2019

On the radical of a Hecke-Kiselman algebra

The Hecke-Kiselman algebra of a finite oriented graph $Θ$ over a field $K$ is studied. If $Θ$ is an oriented cycle, it is shown that the algebra is semiprime and its central localization is a finite direct product of matrix algebras over the field of rational functions $K(x)$. More generally, the radical is described in the case of PI-algebras, and it is shown that it comes from an explicitly described congruence on the underlying Hecke-Kiselman monoid. Moreover, the algebra modulo the radical is again a Hecke-Kiselman algebra and it is a finite module over its center.

math.RA↗

arXiv2019

Combinatorics and structure of Hecke-Kiselman algebras

Hecke-Kiselman monoids $\textrm{HK}_Θ$ and their algebras $K[\textrm{HK}_Θ]$, over a field $K$, associated to finite oriented graphs $Θ$ are studied. In the case $Θ$ is a cycle of length $n\geqslant 3$, a hierarchy of certain unexpected structures of matrix type is discovered within the monoid $C_n=\textrm{HK}_Θ$ and it is used to describe the structure and the properties of the algebra $K[C_n]$. In particular, it is shown that $K[C_n]$ is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand-Kirillov dimension one. This is used to characterize all Noetherian algebras $K[\textrm{HK}_Θ]$ in terms of the graphs $Θ$. The strategy of our approach is based on the crucial role played by submonoids of the form $C_n$ in combinatorics and structure of arbitrary Hecke-Kiselman monoids $\textrm{HK}_Θ$.

math.RA↗