Serge Ovsienko

Fibers of characters in Gelfand-Tsetlin categories

For a class of noncommutative rings, called Galois orders, we study the problem of an extension of characters from a commutative subalgebra. We show that for Galois orders this problem is always solvable in the sense that all characters can be extended, moreover, in finitely many ways, up to isomorphism. These results can be viewed as a noncommutative analogue of liftings of prime ideals in the case of integral extensions of commutative rings. The proposed approach can be applied to the representation theory of many infinite dimensional algebras including universal enveloping algebras of reductive Lie algebras (in particular gln), Yangians and finite W -algebras. As an example we recover the theory of Gelfand-Tsetlin modules for gln.

Submodule Structure of Generalized Verma Modules Induced from Generic Gelfand-Zetlin Modules

For complex Lie algebra sl(n, C) we study the submodule structure of generalized Verma modules induced from generic Gelfand-Zetlin modules over some subalgebra of type sl(k, C). We obtain necessary and sufficient conditions for the existence of a submodule generalizing the Bernstein-Gelfand-Gelfand theorem for Verma modules.

Strongly nilpotent matrices and Gelfand–Zetlin modules

Abstract Let X n be the variety of n × n matrices, which k × k submatrices, formed by the first k rows and columns, are nilpotent for any k =1,…, n . We show, that X n is a complete intersection of dimension ( n −1) n /2 and deduce from it, that every character of the Gelfand–Zetlin subalgebra in U ( gl n ) extends to an irreducible representation of U ( gl n ).

Kostant's Theorem for Special Filtered Algebras

A famous result of Kostants states that the universal enveloping algebra of a semisimple complex Lie algebra is a free module over its center. An analogue of this result is proved for the class of special filtered algebras. This is then applied to show that the restricted Yangian and the universal enveloping algebra of the restricted current algebra, associated with the general linear Lie algebra, are both free over their centers.

Harish-Chandra modules for Yangians

We study Harish-Chandra representations of Yangian for gl(2). We prove an analogue of Kostant theorem showing that resterited Yangians for gl(2) are free modules over certain maximal commutative subalgebras. We also study the categories of generic Harish-Chandra modules, describe their simple modules and indecomposable modules in tame blocks.