Vyacheslav Futorny

University of São Paulo, San Paulo, São Paulo, Brazil

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Publications (85)41.15 Total impact

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    ABSTRACT: We solve the noncommutative Noether's problem for the reflection groups by showing that the skew ?eld of the invariants of the Weyl algebra under the action of any reection group is a Weyl ?field, that is isomorphic to a skew fi?eld of some Weyl algebra over a transcendental extension of the ground fi?eld. We also extend this result to the invariants of the ring of di?fferential operators on any dimensional torus.The results are applied to obtain analogs of the Gelfand-Kirillov Conjecture for Cherednik algebras and Galois algebras.
  • Ben Cox, Vyacheslav Futorny, Kailash C. Misra
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    ABSTRACT: We consider imaginary Verma modules for quantum affine algebra U-q((g) over cap), where (g) over cap has Coxeter-Dynkin diagram of ADE type, and construct Kashiwara type operators and the Kashiwara algebra K-q. We show that a certain quotient N-q(-) of U-q((g) over cap) is a simple K-q-module.
    Journal of Algebra 02/2015; 424:390-415. DOI:10.1016/j.jalgebra.2014.09.025 · 0.60 Impact Factor
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    ABSTRACT: 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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    ABSTRACT: We provide a classification and explicit bases of tableaux of all irreducible generic Gelfand-Tsetlin modules for the Lie algebra gl(n).
    Symmetry Integrability and Geometry Methods and Applications 09/2014; 11. DOI:10.3842/SIGMA.2015.018 · 1.30 Impact Factor
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    ABSTRACT: The classical Gelfand-Tsetlin formulas provide a basis in terms of tableaux and an explicit action of the generators of $\mathfrak{gl} (n)$ for every irreducible finite-dimensional $\mathfrak{gl} (n)$-module. These formulas can be used to define a $\mathfrak{gl} (n)$-module structure on some infinite-dimensional modules - the so-called generic Gelfand-Tsetlin modules. The generic Gelfand-Tsetlin modules are convenient to work with since for every generic tableau there exists a unique irreducible generic Gelfand-Tsetlin module containing this tableau as a basis element. In this paper we initiate the systematic study of a large class of non-generic Gelfand-Tsetlin modules - the class of $1$-singular Gelfand-Tsetlin modules. An explicit tableaux realization and the action of $\mathfrak{gl} (n)$ on these modules is provided using a new construction which we call derivative tableaux. Our construction of $1$-singular modules provides a large family of new irreducible Gelfand-Tsetlin modules of $\mathfrak{gl} (n)$, and is a part of the classification of all such irreducible modules for $n=3$.
  • VYACHESLAV FUTORNY, SERGE OVSIENKO
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    ABSTRACT: We solve the problem of extension of characters of commutative subalgebras in associative (noncommutative) algebras for a class of subrings (Galois rings) in skew group rings. These results can be viewed as a noncom- mutative analogue of liftings of prime ideals in the case of integral extensions of commutative rings. The proposed approach can be applied to the rep- resentation theory of many infinite dimensional algebras including universal enveloping algebras of reductive Lie algebras, Yangians and finite W-algebras. In particular, we advance in the representation theory of gln developing a the- ory of Gelfand-Tsetlin modules. Besides classification results we characterize their categories in the generic case.
    Transactions of the American Mathematical Society 08/2014; 366(8). DOI:10.1090/S0002-9947-2014-05938-2 · 1.10 Impact Factor
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    Vyacheslav Futorny, Dimitar Grantcharov, Renato A. Martins
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    ABSTRACT: We use localization technique to construct new families of irreducible modules of affine Kac-Moody algebras. In particular, localization is applied to the first free field realization of the affine Lie algebra A_1^{(1)} or, equivalently, to imaginary Verma modules.
    Letters in Mathematical Physics 04/2014; 105(4). DOI:10.1007/s11005-015-0752-3 · 2.07 Impact Factor
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    Vyacheslav Futorny, Alexander Molev
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    ABSTRACT: Given a simple Lie algebra $\mathfrak{g}$ and an element $\mu\in\mathfrak{g}^*$, the corresponding shift of argument subalgebra of $\text{S}(\mathfrak{g})$ is Poisson commutative. In the case where $\mu$ is regular, this subalgebra is known to admit a quantization, that is, it can be lifted to a commutative subalgebra of $\text{U}(\mathfrak{g})$. We show that if $\mathfrak{g}$ is of type $A$, then this property extends to arbitrary $\mu$, thus proving a conjecture of Feigin, Frenkel and Toledano Laredo. The proof relies on an explicit construction of generators of the center of the affine vertex algebra at the critical level.
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    ABSTRACT: 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].
    Linear Algebra and its Applications 04/2014; 446:388-420. DOI:10.1016/j.laa.2014.01.016 · 0.98 Impact Factor
  • Yuly Billig, Vyacheslav Futorny
    Journal für die reine und angewandte Mathematik (Crelles Journal) 01/2014; DOI:10.1515/crelle-2014-0059 · 1.30 Impact Factor
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    Ben Cox, Vyacheslav Futorny, Renato Alessandro Martins
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    ABSTRACT: We use the description of the universal central extension of the DJKM algebra $\mathfrak{sl}(2, R)$ where $ R=\mathbb C[t,t^{-1},u\,|\,u^2=t^4-2ct^2+1 ]$ given in earlier work to construct realizations of the DJKM algebra in terms of sums of partial differential operators.
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    Ben Cox, Vyacheslav Futorny, Kailash C. Misra
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    ABSTRACT: We consider imaginary Verma modules for quantum affine algebra $U_q(\hat{\mathfrak{g}})$, where $\hat{\mathfrak{g}}$ is of type 1 i.e. of non-twisted type, and construct Kashiwara type operators and the Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal N_q^-$ of $U_q(\hat{\mathfrak{g}})$ is a simple $\mathcal K_q$-module.
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    Ben Cox, Ilaria Damiani, Vyacheslav Futorny, Kailash C. Misra
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    ABSTRACT: Let $\hat{\mathfrak g}$ be an affine Lie algebra of type 1. We give a PBW basis for the quantum affine algebra $U_q(\hat{\mathfrak g})$ with respect to the triangular decomposition of $\hat{\mathfrak g}$ associated with the imaginary positive root system.
    Journal of Pure and Applied Algebra 07/2013; 219(1). DOI:10.1016/j.jpaa.2014.04.011 · 0.58 Impact Factor
  • P. Bressler, V. Futorny
    The Quarterly Journal of Mathematics 06/2013; 65(2):581-596. DOI:10.1093/qmath/hat036 · 0.59 Impact Factor
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    Yuly Billig, Vyacheslav Futorny
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    ABSTRACT: We give a new conceptual proof of the classification of cuspidal modules for the solenoidal Lie algebra. This classification was originally published by Y.Su. Our proof is based on the theory of modules for the solenoidal Lie algebras that admit a compatible action of the commutative algebra of functions on a torus.
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    Yuly Billig, Vyacheslav Futorny
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    ABSTRACT: We classify all simple $W_n$-modules with finite-dimensional weight spaces. Every such module is either of a highest weight type or is a quotient of a module of tensor fields on a torus, which was conjectured by Eswara Rao. This generalizes the classical result of Mathieu on simple weight modules for the Virasoro algebra. In our proof of the classification we construct a functor from the category of cuspidal $W_n$-modules to the category of $W_n$-modules with a compatible action of the algebra of functions on a torus. We also present a new identity for certain quadratic elements in the universal enveloping algebra of $W_1$, which provides important information about cuspidal $W_1$-modules.
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    Ben Cox, Vyacheslav Futorny, Juan A. Tirao
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    ABSTRACT: We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa in their work on the Landau-Lifshitz equations. Two of the families of polynomials we show satisfy certain forth order linear differential equations, are orthogonal and are not of classical type
    Journal of Differential Equations 10/2012; 255(9). DOI:10.1016/j.jde.2013.07.020 · 1.57 Impact Factor
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    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 subset of 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 go : U -> V such that phi(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.
    Linear Algebra and its Applications 08/2012; 438(5). DOI:10.1016/j.laa.2012.10.038 · 0.98 Impact Factor
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    ABSTRACT: We classify simple weight modules over infinite dimensional Weyl algebras and realize them using the action on certain localizations of the polynomial ring. We describe indecomposable projective and injective weight modules and deduce from this a description of blocks of the category of weight modules by quivers and relations. As a corollary we establish Koszulity for all blocks.
    Proceedings of the American Mathematical Society 07/2012; DOI:10.1090/S0002-9939-2014-12071-5 · 0.63 Impact Factor
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    Vyacheslav Futorny, Jonas T. Hartwig
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    ABSTRACT: It was shown by the first author and Ovsienko that the universal enveloping algebra of $\mathfrak{gl}_N$ is a Galois order, that is, it has a hidden invariant skew group structure. We extend this result to the quantized case and prove that $U_q(\mathfrak{gl}_N)$ is a Galois order over its Gelfand-Tsetlin subalgebra. This leads to a parameterization of finite families of isomorphism classes of irreducible Gelfand-Tsetlin modules for $U_q(\mathfrak{gl}_N)$ by the characters of Gelfand-Tsetlin subalgebra. In particular, any character of the Gelfand-Tsetlin subalgebra extends to an irreducible Gelfand-Tsetlin module over $U_q(\mathfrak{gl}_N)$ and, moreover, extends uniquely when such character is generic. We also obtain a proof of the fact that the Gelfand-Tsetlin subalgebra of $U_q(\mathfrak{gl}_N)$ is maximal commutative, as previously conjectured by Mazorchuk and Turowska.

Publication Stats

348 Citations
41.15 Total Impact Points

Institutions

  • 2000–2014
    • University of São Paulo
      • Institute of Mathematics and Statistics (IME) (São Paulo)
      San Paulo, São Paulo, Brazil
  • 2001
    • Federal University of Minas Gerais
      • Departamento de Matemática
      Cidade de Minas, Minas Gerais, Brazil
    • Utah State University
      Logan, Ohio, United States
  • 1995–2001
    • Kiev Slavonic University
      Kievo, Kyiv City, Ukraine
  • 1994–1995
    • Queen's University
      • Department of Mathematics & Statistics
      Kingston, Ontario, Canada