Jan Okniński

Submonoids of polycyclic-by-finite groups and their algebras

We describe Noetherian semigroup algebras K[S] of submonoids S of polycyclic-by-finite groups over a field K. As an application, we show that these algebras are finitely presented and also that they are Jacobson rings. Next we show that every prime ideal P of K[S] is strongly related to a prime ideal of the group algebra of a subgroup of the quotient group of S via a generalised matrix ring structure on K[S]/P. Applications to the classical Krull dimension, prime spectrum, and irreducible K[S]-modules are given.

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.

Semigroup Algebras and Maximal Orders

We describe contracted semigroup algebras of Malcev nilpotent semigroups that are prime Noethe- rian maximal orders.

Semilocal semigroup rings

In the case of semigroup rings some stronger conditions have been studied. Munn examined the semisimple artinian situation [6]. Zelmanov showed that if K[G] is artinian then G must be finite [11]. The purpose of the present paper is to characterize semilocal semigroup rings K[G] be means of the properties of the semigroup G. It is done in the cases listed in Theorem A. Fundamental definitions and properties of semigroups and group rings may be found in [1, 10]. In what follows K will be a field and G a semigroup. If G contains a unity then we shall denote by G, the subgroup of invertible elements in G and put Go= G\Gt. If G has no unity, then Go = G. Let us notice that if G contains a unity and K[G] is semilocal, then K[G] is von Neumann finite [2] and so Go is an ideal in G. The set of idempotents of G will be denoted by E(G). If A is a ring /(A) will denote the Jacobson radical of A. For a semilocal ring A we use nA for the length of A-module AU(A). The starting point for our considerations is the following result.

Chinese Algebras of Rank 3

The structure of the algebra K[M] of the Chinese monoid M of rank 3 over a field K is studied. The minimal prime ideals are described and the classical Krull dimension is computed. It follows that every minimal prime ideal is determined by a homogeneous congruence on M. Moreover, the prime radical is nilpotent and equal to the Jacobson radical. This ideal is not determined by a congruence on M.

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

Abstract 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 [38, Example 3.9], who first gave a counterexample to Gateva-Ivanova’s Strong Conjecture [19, Strong Conjecture 2.28 (I)]. 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 [21, Open Questions 6.13 (II)(4)]. 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.

The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition

We consider algebras over a field K presented by generators x 1 ,...,x n and subject to ( n 2 ) square-free relations of the form x i x j = x k x l with every monomial x i x j , i≠j, appearing in one of the relations. It is shown that for n > 1 the Gelfand-Kirillov dimension of such an algebra is at least two if the algebra satisfies the so-called cyclic condition. It is known that this dimension is an integer not exceeding n. For n ≥ 4, we construct a family of examples of Gelfand-Kirillov dimension two. We prove that an algebra with the cyclic condition with generators x 1 ,...,x n has Gelfand-Kirillov dimension n if and only if it is of I-type, and this occurs if and only if the multiplicative submonoid generated by x 1 ,...,x n is cancellative.

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.

Generalised Tits alternative for linear semigroups

Abstract It is shown that a finitely generated linear semigroup T ⊆ GL ( n , K ) with no free non-commutative subsemigroups generates a nilpotent-by-finite subgroup of GL ( n , K ). This extends the results of Tits and Rosenblatt on finitely generated linear and finitely generated solvable groups. We use it to derive a ‘generalised Tits alternative’ for an arbitrary linear semigroup S ⊆ M ( n , K ) and to obtain consequences for the structure of the Zariski and strongly π-regular closures of such S .

Gröbner bases for quadratic algebras of skew type

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FM n . of rank n the set of normal forms of elements of S is a regular language in FM n . As one of the key ingredients of the proof, it is shown that an identity of the form x N y N = y N x N holds in S . The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K [ S ] is a finite module over a finitely generated commutative subalgebra of the form K [ A ] for a submonoid A of S .