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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.

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

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.

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

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

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

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.

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

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

Trace functions of the Parafermion vertex operator algebrasOct 10 2018The trace functions for the Parafermion vertex operator algebra associated to any finite dimensional simple Lie algebra $\g$ and any positive integer $k$ are studied and an explicit modular transformation formula of the trace functions is obtained.

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

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

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.

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

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

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

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.

Poisson $λ$-brackets for differential-difference equationsJun 14 2018Jun 18 2018We introduce the notion of a multiplicative Poisson $\lambda$-bracket, which plays the same role in the theory of Hamiltonian differential-difference equations as the usual Poisson $\lambda$-bracket plays in the theory of Hamiltonian PDE. We classify ... 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

Computation of cohomology of Lie conformal and Poisson vertex algebrasMar 28 2019We develop methods for computation of Poisson vertex algebra cohomology. This cohomology is computed for the free bosonic and fermionic Poisson vertex (super)algebras, as well as for the universal affine and Virasoro Poisson vertex algebras. We establish ... More

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

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.

Poisson vertex algebra cohomology and differential Harrison cohomologyJul 16 2019We construct a canonical map from the Poisson vertex algebra cohomology complex to the differential Harrison cohomology complex, which restricts to an isomorphism on the top degree. This is an important step in the computation of Poisson vertex algebra ... 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

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.

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

A new scheme of integrability for (bi)Hamiltonian PDEAug 11 2015Sep 05 2018We 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

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

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

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

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

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

An operadic approach to vertex algebra and Poisson vertex algebra cohomologyJun 22 2018We translate the construction of the chiral operad by Beilinson and Drinfeld to the purely algebraic language of vertex algebras. Consequently, the general construction of a cohomology complex associated to a linear operad produces a vertex algebra cohomology ... 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

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

Irreducible modules over finite simple Lie conformal superalgebras of type KMar 23 2010We construct all finite irreducible modules over Lie conformal superalgebras of type K

Conformal embeddings and simple current extensionsOct 24 2012Feb 18 2014In this paper we investigate the structure of intermediate vertex algebras associated with a maximal conformal embedding of a reductive Lie algebra in a semisimple Lie algebra of classical type.

Local and non-local multiplicative Poisson vertex algebras and differential-difference equationsSep 05 2018We develop the notions of multiplicative Lie conformal and Poisson vertex algebras, local and non-local, and their connections to the theory of integrable differential-difference Hamiltonian equations. We establish relations of these notions to $q$-deformed ... 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

Irreducible Modules over Finite Simple Lie Pseudoalgebras II. Primitive Pseudoalgebras of Type KMar 31 2010One 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. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal ... More

Chiral vs classical operadDec 14 2018We establish an explicit isomorphism between the associated graded of the filtered chiral operad and the classical operad, which is useful for computing the cohomology of vertex algebras.

Classical affine W-algebras and the associated integrable Hamiltonian hierarchies for classical Lie algebrasMay 29 2017Mar 12 2018We prove that any classical affine W-algebra W(g,f), where g is a classical Lie algebra and f is an arbitrary nilpotent element of g, carries an integrable Hamiltonian hierarchy of Lax type equations. This is based on the theories of generalized Adler ... 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

On the structure of quantum vertex algebrasJun 12 2019A definition of a quantum vertex algebra, which is a deformation of a vertex algebra, was proposed by Etingof and Kazhdan in 1998. In a nutshell, a quantum vertex algebra is a braided state-field correspondence which satisfies associativity and braided ... 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

Finite vs. affine W-algebrasNov 16 2005Jan 24 2006In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the lambda-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra Zhu_G ... More

An application of collapsing levels to the representation theory of affine vertex algebrasJan 30 2018Oct 27 2018We discover a large class of simple affine vertex algebras $V_{k} (\mathfrak g)$, associated to basic Lie superalgebras $\mathfrak g$ at non-admissible collapsing levels $k$, having exactly one irreducible $\mathfrak g$-locally finite module in the category ... More

Finite vs infinite decompositions in conformal embeddingsSep 22 2015Apr 06 2016Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras $V_{\mathbf{k}}(\mathfrak g^0)\subset V_{k}(\mathfrak g)$, corresponding to an embedding of a maximal equal rank reductive subalgebra $\mathfrak ... 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

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.

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

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).

Polynomial tau-functions for the multi-component KP hierarchyJan 23 2019In a previous paper we constructed all polynomial tau-functions of the 1-component KP hierarchy, namely, we showed that any such tau-function is obtained from a Schur polynomial $s_\lambda(t)$ by certain shifts of arguments. In the present paper we give ... More

Polynomial tau-functions of BKP and DKP hierarchiesNov 21 2018We construct all polynomial tau-functions of the BKP, DKP and MDKP hierarches.

Equivalence of formulations of the MKP hierarchy and its polynomial tau-functionsJan 09 2018May 09 2018We give 4 formulations of the Modified KP hierarchy and show that they are equivalent. We also discuss the reductions of the MKP hierarchy to the modified $n$-KdV hierarchies. As a byproduct, we find an astonishingly simple explicit description of all ... More

A Lax type operator for quantum finite W-algebrasJul 12 2017Sep 19 2018For a reductive Lie algebra g, its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra W(g,f). We show that for the classical linear Lie algebras gl_N, ... More

Vacuum structure in supersymmetric Yang-Mills theories with any gauge groupFeb 03 1999Apr 30 1999We consider the pure supersymmetric Yang--Mills theories placed on a small 3-dimensional spatial torus with higher orthogonal and exceptional gauge groups. The problem of constructing the quantum vacuum states is reduced to a pure mathematical problem ... More

Normalized Vacuum States in N = 4 Supersymmetric Yang--Mills Quantum Mechanics with Any Gauge GroupAug 13 1999Sep 15 1999We study the question of existence and the number of normalized vacuum states in N = 4 super-Yang-Mills quantum mechanics for any gauge group. The mass deformation method is the simplest and clearest one. It allowed us to calculate the number of normalized ... 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.

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

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.

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 Dynkin gradings in simple Lie algebrasJun 03 2018In this paper we study gradings on simple Lie algebras arising from nilpotent elements. Specifically, we investigate abelian subalgebras which are degree 1 homogeneous with respect to these gradings. We show that for each odd nilpotent element there always ... 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.

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 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

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

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.

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

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.

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