My primary research focus is in commutative algebra and its applications. I am especially interested in understanding the mechanisms underlying various well known asymptotic stability phenomena from representation theory, topology, and combinatorics using the language of commutative algebra. In the following sections I will spend time outlining much of my work in this direction. A particular fo...

For each simple Lie algebra g consider the corresponding affine vertex algebra Vcrit(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A fo...

It will be shown that for 1 < p < 2 the Schatten p-class is isometrically isomor-phic to a subspace of the predual of a von Neumann algebra. Similar results hold for non-commutative L p (N;)-spaces deened by a semi-nite, normal, faithful trace on a von Neumann algebra N. The embeddings rely on a suitable notion of p-stable processes in the non-commutative setting.

This paper introduces a new method in the study of representations of groups into affine algebraic groups. In this work we focus on the study of SLnrepresentations and postpone the consideration of the theory in full to future papers. Let G be a group and let R be a commutative ring with unity. There is a commutative R-algebra Repn(G), called the universal representation algebra, and the univer...

F [D] of Weyl type was defined and studied, where A is a commutative associative algebra with an identity element over a field F of arbitrary characteristic, and F [D] is the polynomial algebra of a commutative derivation subalgebra D of A. The 2-cocycles of a class of A[D] were determined by Su. In the present paper, we determine the 2-cocycles of a class of Lie superalgebras of Weyl type over...

The purpose of this note is to show how recent progress in non-commutative combinatorial algebra gives a new input to Jacobian-related problems in a commutative situation. @ 1997 Elscvier Science B.V. 1991 Math. Subj. Class.: 13B2.5, 13810, 14A05

Effective matrix methods for solving standard linear algebra problems in a commutative domains are discussed. Two of them are new. There are a methods for computing adjoined matrices and solving system of linear equations in a commutative domains.