Leonid Makar-Limanov

The skew field of fractions of the weyl algebra contains a free noncommutative subalgebra

It is proven that the subalgebra of the skew field of fractions of the Weyl algebra which is generated by (pq)-1and (pq)-1 (1-p)-1 where p and q are generators of the Weyl algebra is free.

On groups of automorphisms of a class of surfaces

In this note we describe the group of automorphisms of a commutative algebra with three generatorsx, y andz satisfying a relationxy= P(z), whereP(z) is a polynomial.

On group rings of nilpotent groups

It is shown that ifG is a non-abelian torsion free nilpotent group andF is a field, then the classical skew field of fractionsF(G) of the group ring,F[G] contains a noncommutative free subalgebra.

On Free Subobjects of Skew Fields

In my talk I am going to discuss several questions, which loosely can be stated as follows: does this or that free object appear in a skew field which satisfies certain conditions? Free objects which I have in mind are: free semigroup, free group and free algebra with two generators. It is not reasonable to consider a bigger number of generators, because as is well known, every free object mentioned contains the corresponding free object on a countable number of generators. On the other hand in the case of one generator every object is commutative, and these questions have been considered. It is well known that every skew field which does not coincide with its center contains a free subgroup with one generator (in its multiplicative group). The question about one-generator sub- algebras constitutes the famous Kurosh problem which is still very far from the solution. However, it seems more appropriate to me to consider noncommutative subobjects in the skew field setting.

Newton polytopes of invariants of additive group actions

Abstract It is shown that the vertices of Newton polytopes of invariants of an algebraic group action of the additive group of a field k of arbitrary characteristic on affine n-space over k lie on the coordinate hyperplanes. Furthermore, let E be the set of all edges of these Newton polytopes whose vertices lie in different coordinate hyperplanes. It is shown that one of these polytopes has edges with all directions represented in E.

THE FREIHEITSSATZ AND THE AUTOMORPHISMS OF FREE RIGHT-SYMMETRIC ALGEBRAS

We prove the Freiheitssatz for right-symmetric algebras and the decidability of the word problem for right-symmetric algebras with a single defining relation. We also prove that two generated subalgebras of free right-symmetric algebras are free and automorphisms of two generated free right-symmetric algebras are tame.

Centralizers in free Poisson algebras

We prove an analog of the Bergman Centralizer Theorem for free Poisson algebras over an arbitrary field of characteristic 0. Some open problems are formulated.

Quantization of quadratic Poisson brackets on a polynomial algebra of three variables

Abstract Poisson brackets (P.b.) are the natural initial terms for the deformation quantization of commutative algebras. There is an open problem whether any Poisson bracket on the polynomial algebra of n variables can be quantized. It is known (Poincare-Birkhoff-Witt theorem) that any linear P.b. for all n can be quantized. On the other hand, it is easy to show that in case n = 2 any P.b. is quantizable as well. Quadratic P.b. appear as the initial terms for the quantization of polynomial algebras as quadratic algebras. The problem of the quantization of quadratic P.b. is also open. In the paper we show that in case n = 3 any quadratic P.b. can be quantized. Moreover, the quantization is given as the quotient algebra of tensor algebra of three variables by relations which are similar to those in the Poincare-Birkhoff-Witt theorem. The proof uses a classification of all quadratic Poisson brackets of three variables, which we also give in the paper. In the appendix we give explicit algebraic constructions of the quantized algebras appeared here and show that they are related to algebras of global dimension three considered by M. Artin, W. Schelter, J. Tate, M. Van Den Bergh and other authors from a different point of view.

On subalgebras of the first Weyl skewfield

In this note two subalgebras of the skew field of fractions of the first Weyl algebra are described. One is isomorphic to the group ring of a free group with two generators. Another is maximal in t...

Centralizers in the quantum plane algebra

Dixmier discovered that the centralizers of elements of the first Weyl algebra have some unexpected properties. Sometimes a centralizer is not integrally closed. Also there are cases when the field of fractions of a centralizer is not a purely transcendental field. In this article I am going to discuss what happens if the Weyl algebra is replaced by the quantum plane algebra or a quantum space algebra of any dimension. I became interested in this question after a conversation with L. Small and J. Zhang during a meeting in Taiwan in June of 2001. To my great surprise it turns out that though the centralizers (of non-central elements) are not necessarily integrally closed, the fields of fractions of centralizers of non-constants are always purely transcendental fields of dimension 1 for a “general position” situation.