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 GelfandKirillov Conjecture for Cherednik algebras and Galois algebras.  [Show abstract] [Hide abstract]
ABSTRACT: We consider imaginary Verma modules for quantum affine algebra Uq((g) over cap), where (g) over cap has CoxeterDynkin diagram of ADE type, and construct Kashiwara type operators and the Kashiwara algebra Kq. We show that a certain quotient Nq() of Uq((g) over cap) is a simple Kqmodule.Journal of Algebra 02/2015; 424:390415. DOI:10.1016/j.jalgebra.2014.09.025 · 0.60 Impact Factor  [Show abstract] [Hide abstract]
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 KacMoody 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}.  [Show abstract] [Hide abstract]
ABSTRACT: We provide a classification and explicit bases of tableaux of all irreducible generic GelfandTsetlin 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  [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 $n=3$.  [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). DOI:10.1090/S000299472014059382 · 1.10 Impact Factor  [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.Letters in Mathematical Physics 04/2014; 105(4). DOI:10.1007/s1100501507523 · 2.07 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.  [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. DOI:10.1016/j.laa.2014.01.016 · 0.98 Impact Factor  Journal für die reine und angewandte Mathematik (Crelles Journal) 01/2014; DOI:10.1515/crelle20140059 · 1.30 Impact Factor
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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^42ct^2+1 ]$ given in earlier work to construct realizations of the DJKM algebra in terms of sums of partial differential operators.  [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.  [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; 219(1). DOI:10.1016/j.jpaa.2014.04.011 · 0.58 Impact Factor 
Article: CHIRAL ANOMALY VIA VERTEX ALGEBROIDS
The Quarterly Journal of Mathematics 06/2013; 65(2):581596. DOI:10.1093/qmath/hat036 · 0.59 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.  [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.  [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; 255(9). DOI:10.1016/j.jde.2013.07.020 · 1.57 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: For a finite poset P = {p(1),..., p(t)), we study systems (U1,..., Ut)(U) of subspaces of a unitary space U such that Ui subset of Uj if p(i) < p(j). Two systems (U1,..., Ut)(U) and (V1,..., Vt)(V) are said to be isometric if there exists an isometry go : U > V such that phi(Ui) = Vi. 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  [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; DOI:10.1090/S000299392014120715 · 0.63 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.
Publication Stats
348  Citations  
41.15  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


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
