arXiv subjects

Łukasz Kubat

Publications and source records attributed to Łukasz Kubat.

6 recordsLinked to original sources

arXiv2025

Structure algebras of finite set-theoretic solutions of the Yang--Baxter equation

Quadratic algebras related to some classes of finite left non-degenerate solutions (X,r) of the Yang--Baxter equation have been intensively studied since they are the associative ring-theoretical tool to study solutions. These are the monoid algebras K[M(X,r)] and K[A(X,r)], over a field K, of its structure monoid M(X,r) and left derived structure monoid A(X,r). In case r is bijective (and thus also right non-degenerate) it is known that these algebras are representable (hence PI), left and right Noetherian and have finite Gelfand-Kirillov dimension. Moreover, such algebras are domains (or equivalently prime) if and only if they have finite global dimension, which also is equivalent to r being an involutive map. In this paper we deal with structure algebras of arbitrary finite left non-degenerate solutions (X,r), except for the last section. If (X,r) satisfies additional conditions, such as being bijective, idempotent or left derived, it has been shown in a series of papers that K[M(X,r)] is left Noetherian. In the first part of the paper we show that the algebra K[M(X,r)] always is left Noetherian. Via divisibility by generators, we construct an ideal chain in M(X,r) that has very strong algebraic structural properties on its Rees factors. This allows to obtain characterizations of when the algebras K[M(X,r)] and K[A(X,r)] are right Noetherian. Intricate relationships between ring-theoretical and homological properties of these algebras and properties of the solution (X,r) are proven. Furthermore, we describe the cancellative congruences of A(X,r) and M(X,r) as well as the prime spectrum of K[A(X,r)]. This then leads to an explicit formula for the Gelfand-Kirillov dimension of K[M(X,r)] and it equals the classical Krull dimension.

math.RA↗

arXiv2025

Simple solutions of the Yang-Baxter equation

We study simple set-theoretic solutions of the Yang-Baxter equation that are finite and non-degenerate. Such retractable solutions are fully described and to investigate the irretracble solutions we give a new algebraic method. Our approach includes and extends the work of Joyce for quandles and Castelli for involutive solutions, demonstrating that the simplicity of a solution can be understood through its associated permutation skew left brace. In particular, we show that this skew left brace must have the smallest non-zero ideal, and the quotient by this ideal gives a trivial skew left brace of cyclic type; clearly all simple skew left braces satisfy these assumptions. As an application of our approach we construct and characterise new infinite families of simple solutions that are neither involutive nor quandles. Additionally, we show that our method can be applied to simple skew left braces to generate further families of simple solutions.

math.RA↗

arXiv2022

Finite idempotent set-theoretic solutions of the Yang--Baxter equation

It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies of a group) and some maps $q$ and $φ_x$ on $X$, for $x\in X$. This structure turns out to be a group precisely when the associated structure monoid is cancellative and all the maps $φ_x$ are equal to an automorphism of this group. Equivalently, the structure algebra $K[M(X,r)]$ is right Noetherian, or in characteristic zero it has to be semiprime. The structure algebra always is a left Noetherian representable algebra of Gelfand--Kirillov dimension one. To prove these results it is shown that the structure semigroup $S(X,r)$ has a decomposition in finitely many cancellative semigroups $S_u$ indexed by the diagonal, each $S_u$ has a group of quotients $G_u$ that is finite-by-(infinite cyclic) and the union of these groups carries the structure of a left simple semigroup. The case that $X$ equals the diagonal is fully described by a single permutation on $X$.

math.RA↗

arXiv2019

The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation

For a finite involutive non-degenerate solution $(X,r)$ of the Yang--Baxter equation it is known that the structure monoid $M(X,r)$ is a monoid of I-type, and the structure algebra $K[M(X,r)]$ over a field $K$ share many properties with commutative polynomial algebras, in particular, it is a Noetherian PI-domain that has finite Gelfand--Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid $M(X,r)$ and the algebra $K[M(X,r)]$ is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of $M(X,r)$ as a regular submonoid in the semidirect product $A(X,r)\rtimes\mathrm{Sym}(X)$, where $A(X,r)$ is the structure monoid of the rack solution associated to $(X,r)$, we prove that $K[M(X,r)]$ is a module finite normal extension of a commutative affine subalgebra. In particular, $K[M(X,r)]$ is a Noetherian PI-algebra of finite Gelfand--Kirillov dimension bounded by $|X|$. We also characterize, in ring-theoretical terms of $K[M(X,r)]$, when $(X,r)$ is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of $M(X,r)$. These results allow us to control the prime spectrum of the algebra $K[M(X,r)]$ and to describe the Jacobson radical and prime radical of $K[M(X,r)]$. Finally, we give a matrix-type representation of the algebra $K[M(X,r)]/P$ for each prime ideal $P$ of $K[M(X,r)]$. As a consequence, we show that if $K[M(X,r)]$ is semiprime then there exist finitely many finitely generated abelian-by-finite groups, $G_1,\dotsc,G_m$, each being the group of quotients of a cancellative subsemigroup of $M(X,r)$ such that the algebra $K[M(X,r)]$ embeds into $\mathrm{M}_{v_1}(K[G_1])\times\dotsb\times \mathrm{M}_{v_m}(K[G_m])$.

math.RA↗

arXiv2017

A note on identities in plactic monoids and monoids of upper-triangular tropical matrices

This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank $n$ cannot satisfy a non-trivial identity of length less than or equal to $n$. A new identity is then proven to hold for the monoid of $n \times n$ upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank $3$ into the direct product of two copies of the monoid of $3\times 3$ upper-triangular tropical matrices, giving a new proof that the plactic monoid of rank $3$ satisfies a non-trivial identity.

math.CO↗

arXiv2016

Irreducible representations of the Chinese monoid

All irreducible representations of the Chinese monoid $C_n$, of any rank $n$, over a nondenumerable algebraically closed field $K$, are constructed. It turns out that they have a remarkably simple form and they can be built inductively from irreducible representations of the monoid $C_2$. The proof shows also that every such representation is monomial. Since $C_n$ embeds into the algebra $K[C_n]/J(K[C_n])$, where $J(K[C_n])$ denotes the Jacobson radical of the mooned algebra $K[C_n]$, a new representation of $C_n$ as a subdirect product of the images of $C_n$ in the endomorphism algebras of the constructed simple modules follows.

math.RT↗