Publications (77)32.94 Total impact
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ABSTRACT: We provide a classification and explicit bases of tableaux of all irreducible generic GelfandTsetlin modules for the Lie algebra gl(n).09/2014;  [Show abstract] [Hide abstract]
ABSTRACT: The classical GelfandTsetlin formulas provide a basis in terms of tableaux and an explicit action of the generators of $\mathfrak{gl} (n)$ for every irreducible finitedimensional $\mathfrak{gl} (n)$module. These formulas can be used to define a $\mathfrak{gl} (n)$module structure on some infinitedimensional modules  the socalled generic GelfandTsetlin modules. The generic GelfandTsetlin modules are convenient to work with since for every generic tableau there exists a unique irreducible generic GelfandTsetlin module containing this tableau as a basis element. In this paper we initiate the systematic study of a large class of nongeneric GelfandTsetlin modules  the class of $1$singular GelfandTsetlin 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 GelfandTsetlin modules of $\mathfrak{gl} (n)$, and is a part of the classification of all such irreducible modules for $n=3$.09/2014;  [Show abstract] [Hide abstract]
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 Walgebras. In particular, we advance in the representation theory of gln developing a the ory of GelfandTsetlin modules. Besides classification results we characterize their categories in the generic case.Transactions of the American Mathematical Society 08/2014; 366(8). · 1.02 Impact Factor  [Show abstract] [Hide abstract]
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.04/2014;  [Show abstract] [Hide abstract]
ABSTRACT: We use localization technique to construct new families of irreducible modules of affine KacMoody 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.04/2014;  [Show abstract] [Hide abstract]
ABSTRACT: V.I. Arnold [Russian Math. Surveys 26(2) (1971) 2943] 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) 26702700].Linear Algebra and its Applications 04/2014; 446:388420. · 0.97 Impact Factor 
Article: Solution of a qdifference Noether problem and the quantum GelfandKirillov conjecture for 𝔤𝔩 N
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ABSTRACT: The main result of this paper asserts that the skew field of fractions of the quantum algebra U q (𝔤𝔩 N ) for q∈ℂ not a root of unity is isomorphic to a very explicitly given quantum Weyl field over a purely transcendental field extension of ℂ. The proof is based on reduction to a qdifference analogue of the socalled noncommutative Noether problem which asks for an explicit description of the algebra of invariants in the skew field of fractions of a quantum affine space under the action of a Weyl group. Some additional results on various types of noncommutative invariants are obtained.Reviewer: Volodymyr Mazorchuk (Uppsala)Mathematische Zeitschrift. 01/2014; 276(1).  [Show abstract] [Hide abstract]
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^42ct^2+1 ]$ given in earlier work to construct realizations of the DJKM algebra in terms of sums of partial differential operators.09/2013;  [Show abstract] [Hide abstract]
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 nontwisted 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.08/2013;  [Show abstract] [Hide abstract]
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; · 0.53 Impact Factor  [Show abstract] [Hide abstract]
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.06/2013;  [Show abstract] [Hide abstract]
ABSTRACT: We classify all simple $W_n$modules with finitedimensional 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.04/2013;  [Show abstract] [Hide abstract]
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 LandauLifshitz equations. Two of the families of polynomials we show satisfy certain forth order linear differential equations, are orthogonal and are not of classical typeJournal of Differential Equations 10/2012; · 1.48 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: For a given poset, we consider its representations by systems of subspaces of a unitary space ordered by inclusion. We classify such systems for all posets for which an explicit classification is possible.Linear Algebra and its Applications 08/2012; 438(5). · 0.97 Impact Factor  [Show abstract] [Hide abstract]
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; · 0.61 Impact Factor  [Show abstract] [Hide abstract]
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 GelfandTsetlin subalgebra. This leads to a parameterization of finite families of isomorphism classes of irreducible GelfandTsetlin modules for $U_q(\mathfrak{gl}_N)$ by the characters of GelfandTsetlin subalgebra. In particular, any character of the GelfandTsetlin subalgebra extends to an irreducible GelfandTsetlin 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 GelfandTsetlin subalgebra of $U_q(\mathfrak{gl}_N)$ is maximal commutative, as previously conjectured by Mazorchuk and Turowska.03/2012;  [Show abstract] [Hide abstract]
ABSTRACT: A number of recent papers treated the representation theory of partially ordered sets in unitary spaces with the so called orthoscalar relation. Such theory generalizes the classical theory which studies the representations of partially ordered sets in linear spaces. It happens that the results in the unitary case are wellcorrelated with those in the linear case. The purpose of this article is to shed light on this phenomena.Journal of Physics Conference Series 02/2012; 346(1).  [Show abstract] [Hide abstract]
ABSTRACT: We define the Virasoro algebra action on imaginary Verma modules for affine sl(2) and construct the analogs of KnizhnikZamolodchikov equation in the operator form. Both these results are based on a free field realization of imaginary Verma modules.Letters in Mathematical Physics 12/2011; 102(2). · 2.42 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The goal of this paper is to study the representation theory of a classical infinitedimensional Lie algebra  the Lie algebra of vector fields on an Ndimensional torus for N > 1. The case N=1 gives a famous Virasoro algebra (or its centerless version  the Witt algebra). The algebra of vector fields has an important class of tensor modules parametrized by finitedimensional modules of gl(N). Tensor modules can be used in turn to construct bounded irreducible modules for the vector fields on N+1dimensional torus, which are the central objects of our study. We solve two problems regarding these bounded modules: we construct their free field realizations and determine their characters. To solve these problems we analyze the structure of the irreducible modules for the semidirect product of vector fields with the quotient of 1forms by the differentials of functions. These modules remain irreducible when restricted to the subalgebra of vector fields, unless they belongs to the chiral de Rham complex, introduced by MalikovSchechtmanVaintrob.Transactions of the American Mathematical Society 08/2011; · 1.02 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study $\mathbb Z$graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine KacMoody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function $\phi:\mathbb N\rightarrow \{\pm\}$ we associate a $\phi$highest weight module over the Heisenberg Lie algebra and a $\phi$imaginary Verma module over the affine Lie algebra. We show that any $\phi$imaginary Verma module of nonzero level is irreducible.Journal of Algebra 07/2011; · 0.58 Impact Factor
Publication Stats
263  Citations  
32.94  Total Impact Points  
Top Journals
Institutions

2000–2014

University of São Paulo
 Institute of Mathematics and Statistics (IME) (São Paulo)
San Paulo, São Paulo, Brazil


2006

College of Charleston
 Department of Mathematics
Charleston, South Carolina, United States


2002

University of Leicester
Leiscester, England, United Kingdom


1995–2001

Kiev Slavonic University
Kievo, Kyiv City, Ukraine


1998

St. Lawrence University
Canton, New York, United States


1994

Queen's University
 Department of Mathematics & Statistics
Kingston, Ontario, Canada
