Tatiana Gateva-Ivanova

A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation

A bijective map r: X2→X2, where X={x1,…,xn} is a finite set, is called a set-theoretic solution of the Yang–Baxter equation (YBE) if the braid relation r12r23r12=r23r12r23 holds in X3. A nondegenerate involutive solution (X,r) satisfying r(xx)=xx, for all x∈X, is called square-free solution. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions (X,r) and the associated Yang–Baxter algebraic structures—the semigroup S(X,r), the group G(X,r) and the k-algebra A(k,X,r) over a field k, generated by X and with quadratic defining relations naturally arising and uniquely determined by r. We study the properties of the associated Yang–Baxter structures, and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, ...

Noetherian properties of skew polynomial rings with binomial relations

In this work we study standard finitely presented associative algebras over a fixed field K. A restricted class of skew polynomial rings with quadratic relations considered in an earlier work of M. Artin and W. Schelter will be studied. We call them binomial skew polynomial algebras. We establish necessary and sufficient conditions for such an algebra to be a Noetherian domain.

On the noetherianity of some associative finitely presented algebras

Abstract We consider finitely generated associative algebras over a fixed field K of arbitrary characteristic. For such an algebra A we impose some structural restrictions (we call A strictly ordered). We are interested in the implication of strict order on A for its noetherian properties. In particular, we prove that if A is a graded standard finitely presented strictly ordered algebra, then A is left noetherian if and only if it is almost commutative. In this case A has polynomial growth.

On recognisable properties of associative algebras

The paper considers computer algebra in a non-commutative setting. So far, suchinvestigations have been centred on the use of algorithms for equality and of universal properties of algebras. Here, the foundation of all computations is the presentation of the algebra under investigation by a finite number of generators subject to a finite number of defining relations, which satisfy the additional property of forming a Grobner (or standard) basis. Such algebras are called s.f.p.-for standard finite presentation. It is shown that various algebraic properties, such as being finite-dimensional, nilpotent, nil, algebraic are algorithmically recognisable. When the defining relations are words in the generators, this is also shown to be the case, for the properties of being semi-simple, prime, semi-prime, etc.

Set-theoretic solutions of the Yang-Baxter equation, graphs and computations

We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutions of multipermutation level 2.

Quantum Spaces Associated to Multipermutation Solutions of Level Two

We study finite set-theoretic solutions (X,r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over ℂ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra

Global Dimension of Associative Algebras

\mathcal{A}(\mathbb{C},X,r)

Algorithmic determination of the Jacobson radical of monomial algebras

having a q-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of the finite abelian group

Monomial algebras defined by Lyndon words

\mathcal{G}

Noetherian Properties and Growth of some Associative Algebras

of left actions on X. We study the structure of