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A differential analog of a theorem of ChevalleyJan 25 2001In this note a proof of a differential analog of Chevalley's theorem \cite{C} on homomorphism extensions is given. An immediate corollary is a condition of finitenes of extensions of differential algebras and several equivalent definitions of a differentially ... More

Formal distribution algebras and conformal algebrasSep 17 1997Apr 08 1999Conformal algebra is an axiomatic description of the operator product expansion (or rather its Fourier transform) of chiral fields in a conformal field theory. This is a review of recent developments in the subject.

Representations of affine superalgebras and mock theta functions IIFeb 04 2014We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra $\widehat{sl}_{2|1}$ can be modified, using Zwegers' real analytic corrections, to form an $SL_2(\mathbf{Z})$-invariant ... More

Classification of finite simple Lie conformal superalgebrasJun 04 2001The notion of a Lie conformal superalgebra encodes an axiomatic descrption of singular parts of the operator product expansions of chiral fields in conformal field theory. In the paper we give a detailed proof of the classification of all finite simple ... More

Integrable highest weight modules over affine superalgebras and number theoryJul 11 1994In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all finite-dimensional ... More

Quasifinite highest weight modules over the Lie algebra of differential operators on the circleAug 31 1993We classify positive energy representations with finite degeneracies of the Lie algebra $W_{1+\infty}\/$ and construct them in terms of representation theory of the Lie algebra $\hatgl ( \infty R_m )\/$ of infinite matrices with finite number of non-zero ... More

Representations of affine superalgebras and mock theta functionsAug 06 2013We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $\hat{s\ell}_{2|1}$ (resp. $\hat{ps\ell}_{2|2}$) can be modified, using Zwegers' real analytic corrections, to form a modular (resp. $S$-) invariant ... More

Representations of the exceptional Lie superalgebra E(3,6) III: Classification of singular vectorsOct 22 2003We continue the study of irreducible representations of the exceptional Lie superalgebra E(3,6). This is one of the two simple infinite-dimensional Lie superalgebras of vector fields which have a Lie algebra sl(3)\times sl(2)\times gl(1) as the zero degree ... More

Field AlgebrasApr 23 2002May 07 2002A field algebra is a ``non-commutative'' generalization of a vertex algebra. In this paper we develop foundations of the theory of field algebras.

Classification of linearly compact simple Nambu-Poisson algebrasNov 16 2015We introduce the notion of universal odd generalized Poisson superalgebra associated to an associative algebra A, by generalizing a construction made in [5]. By making use of this notion we give a complete classification of simple linearly compact (generalized) ... More

On rationality of W-algebrasNov 14 2007We study the problem of classification of triples ($\mathfrak{g}, f, k$), where $\mathfrak{g}$ is a simple Lie algebra, $f$ its nilpotent element and $k \in \CC$, for which the simple $W$-algebra $W_k (\mathfrak{g}, f)$ is rational.

SUSY Lattice Vertex AlgebrasOct 08 2007We construct and study SUSY lattice vertex algebras. As a simple example, we obtain the simple vertex algebra associated to the vertex algebra $V_c(N3)$ of central charge $c=3/2$, as the SUSY lattice vertex algebra associated to $\mathbb{Z}$ with bilinear ... More

Supersymmetric vertex algebrasMar 27 2006We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields.

Integrable highest weight modules over affine superalgebras and Appell's functionJun 07 2000We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level~1 case. The analysis of this construction shows, in particular, that in the simplest case of the ... More

Classification of linearly compact simple N=6 3-algebrasOct 18 2010Nov 07 2010$N\leq 8$ 3-algebras have recently appeared in N-supersymmetric 3-dimensional Chern-Simons gauge theories. In our previous paper we classified linearly compact simple N = 8 n-algebras for any $n \geq 3$. In the present paper we classify linearly compact ... More

Quantum reduction in the twisted caseApr 20 2004We study the quantum Hamiltonian reduction for affine superalgebras in the twisted case. This leads to a general representation theory of all superconformal algebras, including the twisted ones (like the Ramond algebra). In particular, we find general ... More

Corrections to the book ``Vertex algebras for beginners'', second edition, by Victor KacJan 18 1999These are corrections to the second edition of the book ``Vertex algebras for beginners'', University Lecture Series, 10, American Mathematical Society, Providence, RI, 1998.

Quasifinite representations of W_{\infty}Oct 29 1999We classify the quasifinite highest weight modules over a family of subalgebras W_{\infty}^{n} of the central extension W_{1+\infty} of the Lie algebra of differential operators on the circle consisting of operators of order \geq n. We classify the unitary ... More

Essential variational Poisson cohomologyJun 29 2011In our recent paper [DSK11] we computed the dimension of the variational Poisson cohomology for any quasiconstant coefficient matrix differential operator K of arbitrary order with invertible leading coefficient, provided that the algebra of differential ... More

Non-local Poisson structures and applications to the theory of integrable systemsFeb 01 2013Jul 17 2013We develop a rigorous theory of non-local Poisson structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the Lenard-Magri scheme of integrability to a pair of compatible ... More

Solitons in Affine and Permutation OrbifoldsDec 31 2003Feb 12 2004We consider properties of solitons in general orbifolds in the algebraic quantum field theory framework and constructions of solitons in affine and permutation orbifolds. Under general conditions we show that our construction gives all the twisted representations ... More

Affine orbifolds and rational conformal field theory extensions of W_{1+infinity}Dec 07 1996Dec 15 1996Chiral orbifold models are defined as gauge field theories with a finite gauge group $\Gamma$. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its ... More

Generalized Spencer Cohomology and filtered Deformations of Z-graded Lie SuperalgebrasMay 07 1998Mar 30 1999In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine ... More

Non-local Poisson structures and applications to the theory of integrable systems IINov 11 2012We develop further the Lenard-Magri scheme of integrability for a pair of compatible non-local Poisson structures, which we discussed in Part I. We apply this scheme to several such pairs, proving thereby integrability of various evolution equations, ... More

Non-local Hamiltonian structures and applications to the theory of integrable systems IOct 05 2012We develop a rigorous theory of non-local Hamiltonian structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the Lenard-Magri scheme of integrability to a pair of compatible ... More

Theory of Finite PseudoalgebrasJul 19 2000Mar 16 2001Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional'' analogues of ... More

A new approach to the Lenard-Magri scheme of integrabilityMar 14 2013We develop a new approach to the Lenard-Magri scheme of integrability of bi-Hamiltonian PDE's, when one of the Poisson structures is a strongly skew-adjoint differential operator.

Dirac reduction for Poisson vertex algebrasJun 27 2013Jul 23 2014We construct an analogue of Dirac's reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac's reduction of an arbitrary non-local Poisson structure. We apply this construction ... More

Quantum Reduction for Affine SuperalgebrasFeb 07 2003We extend the homological method of quantization of generalized Drinfeld--Sokolov reductions to affine superalgebras. This leads, in particular, to a unified representation theory of superconformal algebras.

Some remarks on non-commutative principal ideal ringsMay 02 2013We prove some algebraic results on the ring of matrix differential operators over a differential field in the generality of non-commutative principal ideal rings. These results are used in the theory of non-local Poisson structures.

A new scheme of integrability for (bi)Hamiltonian PDEAug 11 2015We develop a new method for constructing integrable Hamiltonian hierarchies of Lax type equations, which combines the fractional powers technique of Gelfand and Dickey, and the classical Hamiltonian reduction technique of Drinfeld and Sokolov. The method ... More

Double Poisson vertex algebras and non-commutative Hamiltonian equationsOct 13 2014May 23 2015We develop the formalism of double Poisson vertex algebras (local and non-local) aimed at the study of non-commutative Hamiltionan PDEs. This is a generalization of the theory of double Poisson algebras, developed by Van den Bergh, which is used in the ... More

Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebrasJan 09 2014Jan 16 2014We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th KdV hierarchy, ... More

Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotentsJun 07 2013Jul 23 2014We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable ... More

Infinite dimensional primitive linearly compact Lie superalgebrasNov 16 2005We classify open maximal subalgebras of all infinite-dimensional linearly compact simple Lie superalgebras. This is applied to the classification of infinite-dimensional Lie superalgebras of vector fields, acting transitively and primitively in a formal ... More

The stability of vacua in two-dimensional gauge theoryOct 31 2000We discuss the stability of vacua in two-dimensional gauge theory for any simple, simply connected gauge group. Making use of the representation of a vacuum in terms of a Wilson line at infinity, we determine which vacua are stable against pair production ... More

Quantum Reduction and Representation Theory of Superconformal AlgebrasApr 07 2003Nov 16 2004We study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction. As an application, we obtain in a unified way free field realizations and determinant formulas for all superconformal algebras. ... More

On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebrasMay 12 2003Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas, Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie algebras, and checked it in the classical cases. We prove the conjecture for type G_2, and also verify ... More

Representations of simple finite Lie conformal superalgebras of type W and SMar 23 2010May 11 2010We construct all finite irreducible modules over Lie conformal superalgebras of type W and S.

Classical W-algebras and generalized Drinfeld-Sokolov bi-Hamiltonian systems within the theory of Poisson vertex algebrasJul 26 2012Aug 12 2013We provide a description of the Drinfeld-Sokolov Hamiltonian reduction for the construction of classical W-algebras within the framework of Poisson vertex algebras. In this context, the gauge group action on the phase space is translated in terms of (the ... More

On integrability of some bi-Hamiltonian two field systems of PDEMay 06 2014We continue the study of integrability of bi-Hamiltonian systems with a compatible pair of local Poisson structures (H_0,H_1), where H_0 is a strongly skew-adjoint operator. This is applied to the construction of some new two field integrable systems ... More

Integrability of Dirac reduced bi-Hamiltonian equationsJan 23 2014Jul 23 2014First, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three hierarchies of bi-Hamiltonian ... More

Singular degree of a rational matrix pseudodifferential operatorAug 12 2013In our previous work we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H=A/B, where A and B are matrix differential operators, and B is non-degenerate of minimal possible degree deg(B). In the present paper ... More

Rational matrix pseudodifferential operatorsJun 19 2012The skewfield K(d) of rational pseudodifferential operators over a differential field K is the skewfield of fractions of the algebra of differential operators K[d]. In our previous paper we showed that any H from K(d) has a minimal fractional decomposition ... More

Classical W-algebras for gl_N and associated integrable Hamiltonian hierarchiesSep 23 2015Jun 02 2016We apply the new method for constructing integrable Hamiltonian hierarchies of Lax type equations developed in our previous paper, to show that all W-algebras W(gl_N,f) carry such a hierarchy. As an application, we show that all vector constrained KP ... More

Characters of (relatively) integrable modules over affine Lie superlagebrasJun 26 2014In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules $L$ over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras $\mathfrak{g}$. The problems consists of ... More

Freely generated vertex algebras and non-linear Lie conformal algebrasDec 15 2003Dec 16 2003We introduce the notion of a non--linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a non--linear ... More

On integral representations of q-gamma and q-beta functionsFeb 04 2003We study q-integral representations of the q-gamma and the q-beta functions. This study leads to a very interesting q-constant. As an application of these integral representations, we obtain a simple conceptual proof of a family of identities for Jacobi ... More

Lie conformal algebra cohomology and the variational complexDec 29 2008We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra ... More

$Γ$-Conformal AlgebrasSep 01 1997Sep 02 1997$\Gamma$-conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points. We classify these algebras and their representations in terms of Lie algebras and their representations ... More

Conformal ModulesJun 23 1997Sep 12 1997In this paper we study a class of modules over infinite-dimensional Lie (super)algebras, which we call conformal modules. In particular we classify and construct explicitly all irreducible conformal modules over the Virasoro and the N=1 Neveu-Schwarz ... More

Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural resultsFeb 15 2016Apr 17 2016We find all values of $k\in \mathbb C$, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra $W_k(\mathfrak g,\theta)$ is conformal, where $\mathfrak g$ is a basic simple Lie superalgebra and $-\theta$ its minimal ... More

Kostant's pair of Lie type and conformal embeddingsFeb 08 2018We deal with some aspects of the theory of conformal embeddings of affine vertex algebras, providing a new proof of the Symmetric Space Theorem and a criterion for conformal embeddings of equal rank subalgebras. We finally study some examples of embeddings ... More

Differential Conformal Superalgebras and their FormsMay 28 2008We introduce the formalism of differential conformal superalgebras, which we show leads to the "correct" automorphism group functor and accompanying descent theory in the conformal setting. As an application, we classify forms of N=2 and N=4 conformal ... More

Extensions of conformal modulesSep 12 1997Jan 24 2000In this paper we classify extensions between irreducible finite conformal modules over the Virasoro algebra, over the current algebras and over their semidirect sums.

On classification of Poisson vertex algebrasApr 29 2010Mar 22 2011We describe a conjectural classification of Poisson vertex algebras of CFT type and of Poisson vertex algebras in one differential variable (= scalar Hamiltonian operators).

Classification of infinite-dimensional simple groups of supersymmetries and quantum field theoryDec 30 1999Mar 28 2001Talk given at the conference "Visions in Mathematics toward the year 2000", August 1999, Tel-Aviv.

The idea of localitySep 02 1997This is a review of recent results on conformal (super)algebras. It may be viewed as an amplification of my Wigner medal acceptance speech (given in July 1996 in Goslar, Germany) reproduced in the introduction.

Classification of supersymmetriesFeb 07 2003Dec 01 2002In the first part of my talk I will explain a solution to the extension of Lie's problem on classification of "local continuous transformation groups of a finite-dimensional manifold" to the case of supermanifolds. (More precisely, the problem is to classify ... More

Irreducible Modules over Finite Simple Lie Pseudoalgebras I. Primitive Pseudoalgebras of Type W and SOct 07 2004Jun 20 2005One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra [K]. A Lie pseudoalgebra is a generalization of the notion of a Lie ... More

On the classification of subalgebras of Cend_N and gc_NMar 13 2002The problem of classification of infinite subalgebras of Cend_N and of gc_N that acts irreducibly on $\Bbb C[\partial]^N$ is discussed in this paper.

Complexes of modules over exceptional Lie superalgebras $E (3,8)$ and $E (5,10)$Dec 12 2001In this paper complexes of generalized Verma modules over the infinite-dimensional exceptional Lie superalgebras $E (3,8)$ and $E(5,10)$ are constructed and studied.

Representations of affine superalgebras and mock theta functions IIIMay 05 2015We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ \mathfrak{g}. $ For this we develop a several step modification process of multivariable ... More

Algebraic vs physical N=6 3-algebrasSep 30 2013In our previous paper we classified linearly compact algebraic simple N=6 3-algebras. In the present paper we classify their "physical" counterparts, which actually appear in the N=6 supersymmetric 3-dimensional Chern-Simons theories.

Vertex Operator Superalgebras and Their RepresentationsDec 09 1993After giving some definitions for vertex operator SUPERalgebras and their modules, we construct an associative algebra corresponding to any vertex operator superalgebra, such that the representations of the vertex operator algebra are in one-to-one correspondence ... More

Classification of simple linearly compact n-Lie superalgebrasSep 17 2009Jan 15 2010We classify simple linearly compact n-Lie superalgebras with n>2 over a field F of characteristic 0. The classification is based on a bijective correspondence between non-abelian n-Lie superalgebras and transitive Z-graded Lie superalgebras of the form ... More

Classification of linearly compact simple rigid superalgebrasSep 16 2009The notion of an anti-commutative (resp. commutative) rigid superalgebra is a natural generalisation of the notion of a Lie (resp. Jordan) superalgebra. Intuitively rigidity means that small deformations of the product under the structural group produce ... More

A characterization of modified mock theta functionsOct 19 2015We give a characterization of modified (in the sense of Zwegers) mock theta functions, parallel to that of ordinary theta functions. Namely, modified mock theta functions are characterized by their analyticity properties, elliptic transformation properties, ... More

Twisted Modules over Lattice Vertex AlgebrasFeb 19 2004Mar 31 2004For any integral lattice $Q$, one can construct a vertex algebra $V_Q$ called a lattice vertex algebra. If $\sigma$ is an automorphism of $Q$ of finite order, it can be lifted to an automorphism of $V_Q$. In this paper we classify the irreducible $\sigma$-twisted ... More

Generalized Vertex AlgebrasFeb 04 2006We give a short introduction to generalized vertex algebras, using the notion of polylocal fields. We construct a generalized vertex algebra associated to a vector space h with a symmetric bilinear form. It contains as subalgebras all lattice vertex algebras ... More

Simple Jordan conformal superalgebrasJan 04 2008May 06 2008We classify simple finite Jordan conformal superalgebras and establish preliminary results for the classification of simple finite Jordan pseudoalgebras.

Representations of the Exceptional Lie superalgebra $E(3,6): II. Four series of degenerate modulesDec 28 2000Four $\ZZ_+$-bigraded complexes with the action of the exceptional infinite-dimensional Lie superalgebra E(3,6) are constructed. We show that all the images and cokernels and all but three kernels of the differentials are irreducible E(3,6)-modules. This ... More

Representations of the Exceptional Lie superalgebra E(3,6): I. Degeneracy conditionsDec 28 2000Recently one of the authors obtained a classification of simple infinite-dimensional Lie superalgebras of vector fields which extends the well-known classification of E. Cartan in the Lie algebra case. The list consists of many series defined by simple ... More

Finite dimensional representations of quantum affine algebras at roots of unityOct 25 1994Nov 11 1994We describe explicitly the canonical map $\chi:$ Spec $\ue(\a{g})\ \rightarrow \ $Spec $\ze$, where $\ue(\a{g})$ is a quantum loop algebra at an odd root of unity $\ve$. Here $\ze$ is the center of $\ue(\a{g})$ and Spec $R$ stands for the set of all finite--dimensional ... More

Classification of linearly compact simple Jordan and generalized Poisson superalgebrasAug 15 2006We classify all linearly compact simple Jordan superalgebras over an algebraically closed field of characteristic zero. As a corollary, we deduce the classification of all linearly compact unital simple generalized Poisson superalgebras.

Automorphisms and forms of simple infinite-dimensional linearly compact Lie superalgebrasJan 12 2006We describe the group of continuous automorphisms of all simple infinite-dimensional linearly compact Lie superalgebras and use it in order to classify F-forms of these superalgebras over any field F of characteristic zero.

Introduction to vertex algebras, Poisson vertex algebras, and integrable Hamiltonian PDEDec 02 2015These lectures were given in Session 1: "Vertex algebras, W-algebras, and applications" of INdAM Intensive research period "Perspectives in Lie Theory" at the Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy, December 9, 2014 -- February 28, ... More

Subalgebras of $\gc_N$ and Jacobi polynomialsDec 13 2001We classify the subalgebras of the general Lie conformal algebra $\gc_N$ that act irreducibly on $\C[\partial]^N$ and that are normalized by the $\operatorname{sl}_2$--part of a Virasoro element. The problem turns out to be closely related to classical ... More

The variational Poisson cohomologyJun 01 2011Feb 06 2013It is well known that the validity of the so called Lenard-Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian ... More

Structure theory of finite Lie conformal superalgebrasFeb 12 2004We develop structure theory of finite Lie conformal superalgebras.

Elliptic scaling functions as compactly supported multivariate analogs of the B-splinesNov 05 2013In the paper, we present a family of multivariate compactly supported scaling functions, which we call as elliptic scaling functions. The elliptic scaling functions are the convolution of elliptic splines, which correspond to homogeneous elliptic differential ... More

Rotation properties of isotropic dilation matricesNov 07 2013In the paper, we consider bivariate isotropic dilation matrices that are similar (up to constant factors) to rotation matrices; and we show that, in this case, the two-scale relation can be considered also as the relation between not only dilated but ... More

Polynomial spaces reproduced by elliptic scaling functionsOct 31 2013The Strang-Fix conditions are necessary and sufficient to reproduce spaces of algebraic polynomials up to some degree by integer shifts of compactly supported functions. W. Dahmen and Ch. Micchelli (Linear Algebra Appl. 52/3:217-234, 1983) introduced ... More

A degenerate Hopf bifurcation in retarded functional differential equations, and applications to endemic bubblesFeb 24 2015Jul 17 2015In this paper, we study degenerate Hopf bifurcations in a class of parametrized retarded functional differential equations. Specifically, we are interested in the case where the eigenvalue crossing condition of the classical Hopf bifurcation theorem is ... More

On simplicity of vacuum modulesMay 31 2006Jun 14 2006We find necessary and sufficient conditions of irreducibility of vacuum modules over affine Lie algebras and superalgebras. From this we derive conditions of simplicity of minimal W-algebras. Moreover, in the case of Virasoro and Neveu-Schwarz algebras ... More

On complete reducibility for infinite-dimensional Lie algebrasMay 06 2009Sep 04 2011In this paper we study the complete reducibility of representations of infinite-dimensional Lie algebras from the perspective of the representation theory of vertex algebras.

Finite growth representations of infinite Lie conformal algebrasOct 10 2002We classify all finite growth representations of all infinite rank subalgebras of the Lie conformal algebra gc_1 that contain a Virasoro subalgebra.

Cohomology of Conformal AlgebrasMar 07 1998Nov 25 1998Conformal algebra is an axiomatic description of the operator product expansion of chiral fields in conformal field theory. On the other hand, it is an adequate tool for the study of infinite-dimensional Lie algebras satisfying the locality property. ... More

Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairsFeb 18 2011Jan 10 2012We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie ... More

On the Kernel of the affine Dirac operatorApr 22 2008Sep 14 2009Let L be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form, \sigma an elliptic automorphism of L leaving the form invariant, and A a \sigma-invariant reductive subalgebra of L, such that the restriction of the form ... More

Decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of Z_2-graded Lie algebrasJun 15 2005Jan 23 2006We give uniform formulas for the branching rules of level 1 modules over orthogonal affine Lie algebras for all conformal pairs associated to symmetric spaces. We also provide a combinatorial intepretation of these formulas in terms of certain abelian ... More

Quasifinite representations of classical Lie subalgebras of W_{1+infty}Jan 29 1998We show that there are precisely two, up to conjugation, anti-involutions sigma_{\pm} of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of ... More

Representation theory of the vertex algebra $W_{1 + \infty}$Dec 18 1995In our paper~\cite{KR} we began a systematic study of representations of the universal central extension $\widehat{\Cal D}\/$ of the Lie algebra of differential operators on the circle. This study was continued in the paper~\cite{FKRW} in the framework ... More

Some quantum analogues of solvable Lie groupsAug 27 1993In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras of twisted polynomials ... More

Finite W-algebras for gl_NMay 10 2016We study the quantum finite W-algebras W(gl_N,f), associated to the Lie algebra gl_N, and its arbitrary nilpotent element f. We construct for such an algebra an r_1 x r_1 matrix L(z) of Yangian type, where r_1 is the number of maximal parts of the partition ... More

Classification of finite irreducible modules over the Lie conformal superalgebra CK6Oct 18 2011We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK6, for which E(1, 6) is the annihilation ... More

Structure of classical (finite and affine) W-algebrasApr 02 2014First, we derive an explicit formula for the Poisson bracket of the classical finite W-algebra W^{fin}(g,f), the algebra of polynomial functions on the Slodowy slice associated to a simple Lie algebra g and its nilpotent element f. On the other hand, ... More

Some algebraic properties of differential operatorsJan 10 2012First, we study the subskewfield of rational pseudodifferential operators over a differential field K generated in the skewfield of pseudodifferential operators over K by the subalgebra of all differential operators. Second, we show that the Dieudonne' ... More

Poisson vertex algebras in the theory of Hamiltonian equationsJul 07 2009Dec 10 2009We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible ... More

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine settingApr 25 2007Oct 02 2007We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting ... More