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Vyacheslav Futorny

Publications and source records attributed to Vyacheslav Futorny.

At least 37 records · Page 2Linked to original sources
arXiv2023

Representations of Smith algebras which are free over the Cartan subalgebra

In this paper, we study the category of modules over the Smith algebra which are free of finite rank over the unital polynomial subalgebra generated by the Cartan element $h$ and obtain families of such simple modules of arbitrary rank. In the case of rank one we obtain a full description of the isomorphism classes, a simplicity criterion, and an algorithm to produce all composition series. We show that all such modules have finite length and describe the composition factors and their multiplicity.

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arXiv2023

Crystal bases for reduced imaginary Verma modules of untwisted quantum affine algebras

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras $U_q(\hat{\g})$ and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra $\mathcal K_q$ constructed in earlier work by Ben Cox and two of the authors. We prove the existence of the imaginary crystal base for any object in a suitable category $\mc{O}^q_{red,im}$ containing the reduced imaginary Verma modules for $U_q(\hat{\g})$.

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arXiv2023

The Category of reduced imaginary Verma modules

For an arbitrary affine Lie algebra we study an analog of the category O for the natural Borel subalgebra and zero central charge. We show that such category is semisimple having the reduced imaginary Verma modules as its simple objects. This generalizes the result of Cox, Futorny, Misra in the case of affine sl(2).

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arXiv2023

A Jordan-Schwinger Variant of the Spectral Theorem for Linear Operators

In this paper we show variant of the spectral theorem using an algebraic Jordan-Schwinger map. The advantage of this approach is that we don't have restriction of normality on the class of operators we consider. On the other side, we have the restriction that the class of operators we consider should be of weighted Hilbert-Schmidt class.

math.FA↗
arXiv2023

Stretching maps for tensors

We consider an algebra of even-order square tensors and introduce a stretching map which allows us to represent tensors as matrices. The stretching map could be understood as a generalized matricization. It conserves algebraic properties of the tensors. In the same time, we don't necessarily assume injectivity of the stretching map. Dropping the injectivity condition allows us to construct examples of stretching maps with additional symmetry properties. Furthermore, the noninjectivity leads to the averaging of the tensor and possibly could be used to compress the data.

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arXiv2022

Affine Lie algebras representations induced from Whittaker modules

We use induction from parabolic subalgebras with infinite-dimensional Levi factor to construct new families of irreducible representations for arbitrary Affine Kac-Moody algebra. Our first construction defines a functor from the category of Whittaker modules over the Levi factor of a parabolic subalgebra to the category of modules over the Affine Lie algebra. The second functor sends tensor products of a module over the affine part of the Levi factor (in particular any weight module) and of a Whittaker module over the complement Heisenberg subalgebra to the Affine Lie algebra modules. Both functors preserves irreducibility when the central charge is nonzero.

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arXiv2021

Admissible representations of simple affine vertex algebras

We provide an explicit combinatorial description of highest weights of simple highest weight modules over the simple affine vertex algebra of type A of admissible level k. For admissible simple highest weight modules corresponding to the principal, subregular and maximal parabolic nilpotent orbits we give a realization using the Gelfand-Tsetlin theory, which also allows us to obtain a realization of certain classes of simple admissible sl(2)-induced modules in these orbits. In particular, simple admissible sl(2)-induced modules are fully classified for the principal nilpotent orbit.

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arXiv2021

A Poisson Algebra on the Hida Test Functions and a Quantization using the Cuntz Algebra

In this note we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan-Schwinger (J.-S.) map which has been known and used for a long time by physicists. The difference, comparing to J.-S. map, is that we use generators of Cuntz algebra $\mathcal{O}_{\infty}$ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a "building blocks" instead of creation-annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization i.e. exact conservation of Lie bracket and linearity.

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arXiv2021

Simple modules for Affine vertex algebras in the minimal nilpotent orbit

We explicitly construct, in terms of Gelfand--Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl_{n+1}) in the minimal nilpotent orbit of sl_{n+1}. These representations are quotients of induced modules over the affine Kac-Moody algebra of sl_n+1 and include in particular all admissible simple highest weight modules and all simple modules induced from sl_2. Any such simple module in the minimal nilpotent orbit has bounded weight multiplicities.

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arXiv2021

Holonomic modules for rings of invariant differential operators

We study holonomic modules for the rings of invariant differential operators on affine commutative domains with finite Krull dimension with respect to arbitrary actions of finite groups. We prove the Bernstein inequality for these rings. Our main tool is the filter dimension introduced by Bavula. We extend the results for the invariants of the Weyl algebra with respect to the symplectic action of a finite group, for the rings of invariant differential operators on quotient varieties, and invariants of certain generalized Weyl algebras under the linear actions. We show that the filter dimension of all above mentioned algebras equals 1.

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arXiv2020

Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix

I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A,B) of commuting nilpotent operators on a vector space contains the problem of classifying an arbitrary t-tuple of linear operators. Moreover, it contains the problem of classifying representations of an arbitrary quiver, and so it is considered as hopeless. We give a simple normal form of the matrices of (A,B) if the intersection of kernels of A and B is one-dimensional. We prove that this form is canonical if the Jordan matrix of A is a direct sum of Jordan blocks of the same size and the field is of zero characteristic.

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arXiv2020

Congruence of matrix spaces, matrix tuples, and multilinear maps

Two matrix vector spaces $V,W\subset \mathbb C^{n\times n}$ are said to be equivalent if $SVR=W$ for some nonsingular $S$ and $R$. These spaces are congruent if $R=S^T$. We prove that if all matrices in $V$ and $W$ are symmetric, or all matrices in $V$ and $W$ are skew-symmetric, then $V$ and $W$ are congruent if and only if they are equivalent. Let $F: U\times\dots\times U\to V$ and $G: U'\times\dots\times U'\to V'$ be symmetric or skew-symmetric $k$-linear maps over $\mathbb C$. If there exists a set of linear bijections $φ_1,\dots,φ_k:U\to U'$ and $ψ:V\to V'$ that transforms $F$ to $G$, then there exists such a set with $φ_1=\dots=φ_k$.

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arXiv2020

LD-stability for Goldie rings

The lower transcendence degree, introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable algebras, the lower transcendence degree coincides with the Gelfand-Kirillov dimension. We show that the following algebras are LD-stable and compute their lower transcendence degrees: rings of differential operators of affine domains, universal enveloping algebras of finite dimensional Lie superalgebras, symplectic reflection algebras and their spherical subalgebras, finite $W$-algebras of type $A$, generalized Weyl algebras over Noetherian domain (under a mild condition), some quantum groups. We show that the lower transcendence degree behaves well with respect to the invariants by finite groups, and with respect to the Morita equivalence. Applications of these results are given.

math.RA↗
arXiv2020

Classification of simple strong Harish-Chandra $W(m,n)$-modules

We classify all simple strong Harish-Chandra modules for the Lie superalgebra $W(m,n)$. We show that every such module is either strongly cuspidal or a module of the highest weight type. We construct tensor modules for $W(m,n)$, which are parametrized by simple finite-dimensional $gl(m,n)$-modules and show that every simple strongly cuspidal $W(m,n)$-module is a quotient of a tensor module. Finally, we realize modules of the highest weight type as simple quotients of the generalized Verma modules induced from tensor modules for $W(m-1,n)$.

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arXiv2020

Gelfand-Tsetlin modules for gl(n,m)

We address the problem of classifying of irreducible Gelfand-Tsetlin modules for gl(m|n) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. We also give an explicit tableaux construction and the irreducibility criterion for the class of quasi typical and quasi covariant Gelfand-Tsetlin modules which includes all essentially typical and covariant tensor finite dimensional modules. In the quasi typical case new irreducible representations are infinite dimensional gl(m|n)-modules which are isomorphic to the parabolically induced (Kac) modules.

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arXiv2020

Positive energy representations of affine vertex algebras

We construct new families of positive energy representations of affine vertex algebras together with their free field realizations by using localization technique. We introduce the twisting functor T_αon the category of modules over affine Kac--Moody algebra \widehat{g}_κof level κfor any positive root αof g, and the Wakimoto functor from a certain category of g-modules to the category of smooth \widehat{g}_κ-modules. These two functors commute and the image of the Wakimoto functor consists of relaxed Wakimoto \widehat{g}_κ-modules. In particular, applying the twisting functor T_αto the relaxed Wakimoto \widehat{g}_κ-module whose top degree component is isomorphic to the Verma g-module M^g_b(λ), we obtain the relaxed Wakimoto \widehat{g}_κ-module whose top degree component is isomorphic to the α-Gelfand--Tsetlin g-module W^g_b(λ, α). We show that the relaxed Verma module and relaxed Wakimoto module whose top degree components are such α-Gelfand--Tsetlin modules, are isomorphic generically. This is an analogue of the result of E.Frenkel for Wakimoto modules both for critical and non-critical level. For a parabolic subalgebra p of g we construct a large family of admissible g-modules as images under the twisting functor of generalized Verma modules induced from p. In this way, we obtain new simple positive energy representations of simple affine vertex algebras.

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arXiv2020

Classification of irreducible Gelfand-Tsetlin modules of sl(3)

We provide a classification and an explicit realization of all irreducible Gelfand-Tsetlin modules of the complex Lie algebra sl(3). The realization of these modules uses regular and derivative Gelfand-Tsetlin tableaux. In particular, we list all simple Gelfand-Tsetlin sl(3)-modules with infinite-dimensional weight spaces. Also, we express all simple Gelfand-Tsetlin sl(3)-modules as subquotionets of localized Gelfand-Tsetlin E_{21}-injective modules.

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