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V. Futorny

Publications and source records attributed to V. Futorny.

10 recordsLinked to original sources
arXiv2026

Graded identities of the first Weyl algebra and its generalizations

We study the graded polynomial identities of the first Weyl algebra $W_1$ over an infinite field. The algebra $W_1$ satisfies no ordinary polynomial identities in characteristic 0. It admits a natural grading by the infinite cyclic group $\mathbb{Z}$. We construct a basis of the $\mathbb{Z}$-graded identities of $W_1$, which consists of a single identity. It expresses the fact that the degree 0 component in the grading is commutative. It is also well known that if the characteristic of the base field is $p>2$, then $W_1$ satisfies the same identities as the full matrix algebra of order $p$. In this situation, we describe the $\mathbb{Z}_p$-graded identities of $W_1$. Afterwards, using various combinatorial and algebraic tools we consider graded identities for various types of algebras generalizing the Weyl algebras. For example, we show that $\mathbb{Z}$-graded Galois rings in characteristic 0 satisfy the same graded identities as $W_1$ when they embed in a shift operator algebra $\mathcal{S}_1$, and as a consequence we obtain that these $\mathbb{Z}$-graded Galois rings are not PI. The same holds for the algebra of differential operators on $1$-dimensional torus. We obtain similar results for generalized Weyl algebras. We also deal with the graded identities for the quantum Weyl algebras and for the quantum plane. It turns out that in the latter case and when $q$ is the $\ell$-th primitive root of unity, one is led to study gradings by the group $\mathbb{Z}_\ell\times \mathbb{Z}_\ell$. In this case the quantum plane satisfies the same graded identities as the matrix algebra of order $\ell$. Finally we construct a natural $\mathbb{Z}$-grading on the universal enveloping algebra of $\mathfrak{sl}_2$, and prove that its $\mathbb{Z}$-graded identities are the same as those of $W_1$, in characteristic $0$.

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arXiv2019

Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators $I_n$

For the algebra $I_n$ of polynomial integro-differential operators over a field $K$ of characteristic zero, a classification of simple weight and generalized weight (left and right) $I_n$-modules is given. It is proven that the category of weight $I_n$-modules is semisimple. An explicit description of generalized weight $I_n$-modules is given and using it a criterion is obtained for the problem of classification of indecomposable generalized weight $I_n$-modules to be of finite representation type, tame or wild. In the tame case, a classification of indecomposable generalized weight $I_n$-modules is given. In the wild case `natural` tame subcategories are considered with explicit description of indecomposable modules. It is proven that every generalized weight $I_n$-module is a unique sum of absolutely prime modules. For an arbitrary ring $R$, we introduce the concept of {\em absolutely prime} $R$-module (a nonzero $R$-module $M$ is absolutely prime if all nonzero subfactors of $M$ have the same annihilator). It is shown that every indecomposable generalized weight $I_n$-module is equidimensional. A criterion is given for a generalized weight $I_n$-module to be finitely generated.

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arXiv2018

Quantum Linear Galois Algebras

We define a class of quantum linear Galois algebras which include the universal enveloping algebra Uq(gln), the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand-Zetlin algebras of type A, the subalgebras of G-invariants of the quantum affine space, quantum torus for G = G(m, p, n), and of the quantum Weyl algebra for G = Sn. We show that all quantum linear Galois algebras satisfy the quantum Gelfand-Kirillov conjecture. Moreover, it is shown that the the subalgebras of invariants of the quantum affine space and of quantum torus for the reflection groups and of the quantum Weyl algebra for symmetric groups are, in fact, Galois orders over an adequate commutative subalgebras and free as right (left) modules over these subalgebras. In the rank 1 cases the results hold for an arbitrary finite group of automorphisms when the field is C.

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arXiv2015

Weyl modules associated to Kac-Moody Lie algebras

Weyl modules were originally defined for affine Lie algebras by Chari and Pressley in \cite{CP}. In this paper we extend the notion of Weyl modules for a Lie algebra $\mathfrak{g} \otimes A$, where $\mathfrak{g}$ is any Kac-Moody algebra and A is any finitely generated commutative associative algebra with unit over $\mathbb{C}$, and prove a tensor product decomposition theorem generalizing \cite{CP}.

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arXiv2014

Irreducible completely pointed modules of quantum groups of type $A$

We give a classification of all irreducible completely pointed $U_q(\mathfrak{sl}_{n+1})$ modules over a characteristic zero field in which $q$ is not a root of unity. This generalizes the classification result of Benkart, Britten and Lemire in the non quantum case. We also show that any infinite-dimensional irreducible completely pointed $U_q(\mathfrak{sl}_{n+1})$ can be obtained from some irreducible completely pointed module over the quantized Weyl algebra $A_{n+1}^q$.

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arXiv2013

Miniversal deformations of matrices under *congruence and reducing transformations

V.I. Arnold [Russian Math. Surveys 26(2) (1971) 29-43] 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 the authors in [Linear Algebra Appl. 436 (2012) 2670-2700].

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arXiv2007

Gelfand-Tsetlin bases for representations of finite W-algebras and shifted Yangians

Remarkable subalgebras of the Yangian for gl_n called the shifted Yangians were introduced in a recent work by Brundan and Kleshchev in relation to their study of finite W-algebras. In particular, in that work a classification of finite-dimensional irreducible representations of the shifted Yangians and the associated finite W-algebras was given. We construct a class of these representations in an explicit form via bases of Gelfand-Tsetlin type.

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arXiv2005

Verma modules for Yangians

We study the Verma modules M(mu(u)) over the Yangian Y(a) associated with a simple Lie algebra a. We give necessary and sufficient conditions for irreducibility of M(mu(u)). Moreover, regarding the simple quotient L(mu(u)) of M(mu(u)) as an a-module, we give necessary and sufficient conditions for finite-dimensionality of the weight subspaces of L(mu(u)).

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arXiv2001

Imaginary Verma modules for the extended Affine Lie algebra $sl_2(C_q)$

We consider one of the most natural extended affine Lie lagebras, the algebra $sl_2({\mathbb C}_q)$ and begin a theory of its representations. In particular, we study a class of imaginary Verma modules, obtain a criterion of irreducibility and describe their submodule structure in "general position".

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