Vyacheslav Futorny

Miniversal deformations of matrices of bilinear forms

Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simp ...

Categories of Induced Modules and Standardly Stratified Algebras

We construct a generalization of the BGG-category O, whose blocks correspond to standardly stratified algebras. We prove reciprocity formulae in these categories and present two classes of examples.

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.

Irreducible Generic Gelfand-Tsetlin Modules of gl(n)

We provide a classification and explicit bases of tableaux of all irreducible generic Gelfand{Tsetlin modules for the Lie algebra gl(n).

Miniversal deformations of matrices under *congruence and reducing transformations

Arnold (1971) [1] constructed a miniversal deformation of a square complex matrix under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We give miniversal deformations of matrices of sesquilinear forms; that is, of square complex matrices under *congruence, and construct an analytic reducing transformation to a miniversal deformation. Analogous results for matrices under congruence were obtained by Dmytryshyn, Futorny, and Sergeichuk (2012) [11].

Systems of subspaces of a unitary space

Abstract For a finite poset P = { p 1 , … , p t } , we study systems ( U 1 , … , U t ) U of subspaces of a unitary space U such that U i ⊆ U j if p i ≺ p j . Two systems ( U 1 , … , U t ) U and ( V 1 , … , V t ) V are said to be isometric if there exists an isometry φ : U → V such that φ ( U i ) = V i . We classify such systems up to isometry if P is a semichain. We prove that the problem of their classification is unitarily wild if P is not a semichain. A classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space, which is hopeless in a certain sense.

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.

Highest weight categories of Lie algebra modules

Abstract We study a certain class of categories of Lie algebra modules which include the well-known categories O and O S . We show that all these categories are highest weight categories.

Multiparameter twisted Weyl algebras

Abstract We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordanʼs simple localization of the multiparameter quantized Weyl algebra and Hayashiʼs q -analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter twisted Weyl algebras.

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.