arXiv subjects

Jan Okninski

Publications and source records attributed to Jan Okninski.

13 recordsLinked to original sources

arXiv2025

New classes of IYB groups

It is proven that every finite group of odd order with all Sylow subgroups of nilpotency class at most two is an involutive Yang-Baxter group (IYB group for short), i.e. it admits a structure of left brace. It is also proven that every finite solvable group of even order with all Sylow subgroups of nilpotency class at most two and abelian Sylow 2-subgroups is an IYB group. These results contribute to the open problem asking which finite solvable groups are IYB, in particular they generalize a result of Ben David and Ginosar concerned with finite solvable groups with abelian Sylow subgroups. With the same techniques it is proven that every finite solvable group with all Sylow subgroups nilpotent of class at most two is isomorphic to the multiplicative group of a skew left brace of nilpotent type. It is also proven that every finite group with the Sylow tower property is isomorphic to the multiplicative group of a skew left brace of nilpotent type.

math.GR↗

arXiv2024

Simple solutions of the Yang-Baxter equation of cardinality $p^n$

For every prime number p and integer $n>1$, a simple, involutive, non-degenerate set-theoretic solution $(X,r$) of the Yang-Baxter equation of cardinality $|X| = p^n$ is constructed. Furthermore, for every non-(square-free) positive integer m which is not the square of a prime number, a non-simple, indecomposable, irretractable, involutive, non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation of cardinality $|X| = m$ is constructed. A recent question of Castelli on the existence of singular solutions of certain type is also answered affirmatively.

math.QA↗

arXiv2024

New simple solutions of the Yang--Baxter equation and their permutation groups

A new class of indecomposable, irretractable, involutive, non-degenerate set-theoretic solutions of the Yang--Baxter equation is constructed. This class complements the class of such solutions constructed in \cite{CO22} and together they generalize the class of solutions described in \cite[Theorem 4.7{CO21}. Necessary and sufficient conditions are found in order that these new solutions are simple. For a rich subclass of these solutions the structure of their permutation groups, considered as left braces, is determined. In particular, these results answer a question stated in \cite{CO21}. In the finite case, all these solutions have square cardinality. A new class of finite simple solutions of non-square cardinality such that their permutation groups are simple left braces is also constructed.

math.QA↗

arXiv2016

A family of irretractable square-free solutions of the Yang-Baxter equation

A new family of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin, who first gave a counterexample to Gateva-Ivanova's Strong Conjecture. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova's Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova. It is proved that the natural left brace structure on the permutation group of the solutions in this family has trivial socle. Properties of the permutation group and of the structure group associated to these solutions are also investigated. In particular, it is proved that the structure groups of finite solutions in this subfamily are not poly-(infinite cyclic) groups.

math.GR↗

arXiv2012

Braces and the Yang-Baxter equation

Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation are discussed and many consequences are derived. In particular, for each positive integer $n$ a finite square-free multipermutation solution of the Yang-Baxter equation with multipermutation level $n$ and an abelian involutive Yang-Baxter group is constructed. This answers a problem of Gateva-Ivanova and Cameron. It is also proved that finite non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation whose associated involutive Yang-Baxter group is abelian are retractable in the sense of Etingof, Schedler and Soloviev. Earlier the authors proved this with the additional square-free hypothesis on the solutions. Retractability of solutions is also proved for finite square-free non-degenerate involutive set-theoretic solutions associated to a left brace.

math.RA↗

arXiv2011

Structure of Chinese algebras

The structure of the algebra K[M] of the Chinese monoid M over a field K is studied. The minimal prime ideals are described. They are determined by certain homogeneous congruences on M and they are in a one to one correspondence with diagrams of certain special type. There are finitely many such ideals. It is also shown that the prime radical B(K[M]) of K[M] coincides with the Jacobson radical and the monoid M embeds into the algebra K[M]/B(K[M]). A new representation of M as a submonoid of the direct product of finitely many copies of the bicyclic monoid and finitely many copies of the infinite cyclic monoid is derived. Consequently, M satisfies a nontrivial identity.

math.RA↗

arXiv2010

Grobner-Shirshov bases for plactic algebras

A finite Grobner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite Grobner-Shirshov bases associated to the natural degree-lexicographic ordering on the corresponding free algebra. The latter is in contrast with the case of a strongly related class of algebras, called Chinese algebras, recently considered by Chen Yuqun and Qiu Jianjun.

math.RA↗

arXiv2009

Algebras and groups defined by permutation relations of alternating type

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{σ(1)} a_{σ(2)} ... a_{σ(n)}$, where $σ$ runs through $\Alt_{n}$, the alternating group, is considered. The associated group, defined by the same (group) presentation, is described. A description of the radical of the algebra is found. It turns out that the radical is a finitely generated ideal that is nilpotent and it is determined by a congruence on the underlying monoid, defined by the same presentation.

math.RA↗

arXiv2009

Normal domains with monomial presentations

Let A be a finitely generated commutative algebra over a field K with a presentation A=K < X_{1}, ..., X_{n} | R >, where R is a set of monomial relations in the generators X_{1}, ..., X_{n}. So A = K[S], the semigroup algebra of the monoid S=< X_{1}, ..., X_{n} | R >. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

math.RA↗

arXiv2009

Retractability of set theoretic solutions of the Yang-Baxter equation

It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly-${\mathbb Z}$ groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted union is constructed. This answers in the negative another problem of Gateva-Ivanova. The constructed solution is of multipermutation level 3. Retractability of solutions is also proved in the case where the natural generators of the IYB group are cyclic permutations. Moreover, it is shown that such solutions are generalized twisted unions.

math.GR↗

arXiv2007

Semigroup algebras of submonoids of polycyclic-by-finite groups and maximal orders

Necessary and sufficient conditions are given for a prime Noetherian algebra K[S] of a submonoid S of a polycyclic-by-finite group G to be a maximal order. These conditions are entirely in terms of the monoid S. This extends earlier results of Brown concerned with the group ring case and of the authors for the case where K[S] satisfies a polynomial identity.

math.RA↗

arXiv2007

Primes of height one and a class of Noetherian finitely presented algebras

Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group are obtained. This is then applied to characterize a large class of prime Noetherian maximal orders that satisfy a polynomial identity, based on a special class of submonoids of polycyclic-by-finite groups. The main results are illustrated with new constructions of concrete classes of finitely presented algebras of this type.

math.RA↗

arXiv2002

Quadratic algebras of skew type and the underlying semigroups

We consider algebras over a field K defined by a presentation K , where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one of the relations. Certain sufficient conditions for the algebra to be noetherian and PI are determined. For this, we prove more generally that right noetherian algebras of finite Gelfand-Kirillov dimension defined by homogeneous relations satisfy a polynomial identity. The structure of the underlying monoid, defined by the same presentation, is described. This is used to derive information on the prime radical and minimal prime ideals. Some examples are described in detail. Earlier, Etingof, Schedler and Soloviev, Gateva-Ivanova and Van den Bergh, and the authors considered special classes of such algebras in the contexts of noetherian algebras, Grobner bases, finitely generated solvable groups, semigroup algebras, and set theoretic solutions of the Yang-Baxter equation.

math.RA↗