Communications in Mathematical Physics, Feb 1, 1977
This article deals with the structure and representations of Lie superalgebras (Z 2 -graded Lie a... more This article deals with the structure and representations of Lie superalgebras (Z 2 -graded Lie algebras). The central result is a classification of simple Lie superalgebras over IR and C. Remarks. 1) This article is written for physicists. Consequently some proofs are sketched or omitted. The complete proofs are contained in my forthcoming paper "Lie superalgebras" submitted to the Russian journal Uspechi mat. nauk (see ). 2) The partial results of classification of simple Z 2 -graded Lie algebras were independently obtained by Kaplansky, Freund, Djokovic, Pais, Rittenberg, Nahm, Scheunert . Chapter I. Classification of Simple Lie Superalgebras § ί. Superalgebras and Lie Superalgebras 1. Superalgebras. Suppose A = AQ@AJ is a superalgebra. Call subalgebra or ideal of a superalgebra A a Z 2 -graded subalgebra or ideal. Call homomorphism the one that preserves Z 2 -grading. Direct and semidirect sum of superalgebras is defined as usual. The definition of tensor product is different. Suppose A = A 0 @A 1
We classify all linearly compact simple Jordan superalgebras over an algebraically closed field o... more We 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.
We complete the classification of conformal embeddings of a maximally reductive subalgebra k into... more We complete the classification of conformal embeddings of a maximally reductive subalgebra k into a simple Lie algebra g at non-integrable non-critical levels k by dealing with the case when k has rank less than that of g. We describe some remarkable instances of decomposition of the vertex algebra V k (g) as a module for the vertex subalgebra generated by k. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings A1 × A1 ֒→ C3 at level k = −1/2, and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding A1 × D4 ֒→ C8 at level k = −1/2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.
We prove that any unitary highest weight module over a universal minimal quantum affine W-algebra... more We prove that any unitary highest weight module over a universal minimal quantum affine W-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine W-algebras and the list of all their irreducible positive energy modules. We also classify all irreducible highest weight modules for the simple affine vertex algebras in the cases when the associated simple minimal W-algebra is unitary.
We prove that, for a Poisson vertex algebra V, the canonical injective homomorphism of the variat... more We prove that, for a Poisson vertex algebra V, the canonical injective homomorphism of the variational cohomology of V to its classical cohomology is an isomorphism, provided that V, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables. Contents 1. Introduction 1 2. Variational PVA cohomology 4 3. Preliminaries on the symmetric group and on graphs 6 4. Classical PVA cohomology 9 5. The Main Theorem 14 6. Sesquilinear Harrison cohomology 15 7. Relation between symmetric sesquilinear Harrison and classical PVA cohomology complexes 21 8. Vanishing of the sesquilinear Harrison cohomology 24 9. Proof of the Main Theorem 5.2 32 References 33 Key words and phrases. Poisson vertex algebra (PVA), classical operad, classical PVA cohomology, variational PVA cohomology, sesquilinear Hochschild and Harrison cohomology.
We introduce the formalism of differential conformal superalgebras, which we show leads to the "c... more We 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 superalgebras by means of Galois cohomology.
The problem of representing an integer as a sum of squares of integers has had a long history. On... more The problem of representing an integer as a sum of squares of integers has had a long history. One of the first after antiquity was A. Girard who in 1632 conjectured that an odd prime p can be represented as a sum of two squares iff p ≡ 1 mod 4, and P. Fermat in 1641 gave an "irrefutable proof" of this conjecture. The subsequent work on this problem culminated in papers by A.M. Legendre (1798) and C.F. Gauss (1801) who found explicit formulas for the number of representations of an integer as a sum of two squares. C.G. Bachet in 1621 conjectured that any positive integer can be represented as a sum of four squares of integers, and it took efforts of many mathematicians for about 150 years before J.-L. Lagrange gave a proof of this conjecture in 1770.
We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. ... more We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary W-algebra. We show that for a minimal simple W-algebra W k (g, θ/2) this form can be unitary only when its 1 2 Z-grading is compatible with parity, unless W k (g, θ/2) "collapses" to its affine subalgebra.
In this paper we study the complete reducibility of representations of infinitedimensional Lie al... more In this paper we study the complete reducibility of representations of infinitedimensional Lie algebras from the perspective of representation theory of vertex algebras.
Let g be a finite-dimensional semisimple Lie algebra and (• , •) its Killing form, σ an elliptic ... more Let g be a finite-dimensional semisimple Lie algebra and (• , •) its Killing form, σ an elliptic automorphism of g, and a a σ-invariant reductive subalgebra of g, such that the restriction of the form (• , •) to a is non-degenerate. Let L(g, σ) and L(a, σ) be the associated twisted affine Lie algebras and F σ (p) the σ-twisted Clifford module over L(a, σ), associated to the orthocomplement p of a in g. Under suitable hypotheses on σ and a, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight L(g, σ)-module and F σ (p), into irreducible L(a, σ)-submodules. As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space.
We show that the normalized supercharacters of principal admissible modules over the affine Lie s... more We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra s 2|1 (resp. ps 2|2) can be modified, using Zwegers' real analytic corrections, to form a modular (resp. S-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N = 2 (resp. N = 4) superconformal algebras with central charge 3 1 − (2m + 2)/M , where m ∈ Z ≥0 , M ∈ Z ≥2 , gcd(2m + 2, M) = 1 if m > 0 (resp. 6 (m/M − 1), where m ∈ Z ≥1 , M ∈ Z ≥2 , gcd(2m, M) = 1 if m > 1), whose modified characters and supercharacters form a modular invariant family.
In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)al... more In 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 t his cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalg ebras.
We show that there are precisely two, up to conjugation, anti-involutions σ ± of the algebra of d... more We show that there are precisely two, up to conjugation, anti-involutions σ ± of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D ± of the Lie subalgebra of this algebra fixed by −σ ± , and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra C[u]/(u m+1) and its classical Lie subalgebras of B, C and D types. Character formulas for positive primitive representations of D ± (including all the unitary ones) are obtained. We also realize a class of primitive representations of D ± in terms of free fields and establish a number of duality results between these primitive representations and finite-dimensional irreducible representations of finite-dimensional Lie groups and supergroups. We show that the vacuum module V c of D + carries a vertex algebra structure and establish a relationship between V c for c ∈ 1 2 Z and W-algebras.
In the paper we consider the problem of computation of characters of relatively integrable irredu... more In 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 g. The problems consists of two parts. First, it is the reduction of the problem to the g-module F (L), where g is the associated to L integral Lie superalgebra and F (L) is an integrable irreducible highest weight gmodule. Second, it is the computation of characters of integrable highest weight modules. There is a general conjecture concerning the first part, which we check in many cases. As for the second part, we prove in many cases the KW-character formula, provided that the KW-condition holds, including almost all finite-dimensional g-modules when g is basic, and all maximally atypical non-critical integrable g-modules when g is affine with non-zero dual Coxeter number. Contents † Supported in part by BSF Grant No. 711623. ‡ Supported in part by Simons fellowship.
N ≤ 8 3-algebras have recently appeared in N-supersymmetric 3-dimensional Chern-Simons gauge theo... more N ≤ 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 ≥ 3. In the present paper we classify linearly compact simple N = 6 3-algebras, using their correspondence with simple linearly compact Lie superalgebras with a consistent short Z-grading, endowed with a graded conjugation. We also briefly discuss N = 5 3-algebras.
We complete the classification of conformal embeddings of a maximally reductive subalgebra k into... more We complete the classification of conformal embeddings of a maximally reductive subalgebra k into a simple Lie algebra g at non-integrable non-critical levels k by dealing with the case when k has rank less than that of g. We describe some remarkable instances of decomposition of the vertex algebra V k (g) as a module for the vertex subalgebra generated by k. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings A1 × A1 ֒→ C3 at level k = −1/2, and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding A1 × D4 ֒→ C8 at level k = −1/2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.
International Journal of Geometric Methods in Modern Physics, Sep 1, 2006
We describe the group of continous automorphisms of all simple infinitedimensional linearly compa... more We describe the group of continous automorphisms of all simple infinitedimensional linearly compact Lie superalgebras and use it in order to classify F-forms of these superalgebras over any field F of characteristic zero.
We give a short introduction to generalized vertex algebras, using the notion of polylocal fields... more We 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 of rank equal to dim h and all irreducible representations of these vertex algebras.
Communications in Mathematical Physics, Feb 1, 1977
This article deals with the structure and representations of Lie superalgebras (Z 2 -graded Lie a... more This article deals with the structure and representations of Lie superalgebras (Z 2 -graded Lie algebras). The central result is a classification of simple Lie superalgebras over IR and C. Remarks. 1) This article is written for physicists. Consequently some proofs are sketched or omitted. The complete proofs are contained in my forthcoming paper "Lie superalgebras" submitted to the Russian journal Uspechi mat. nauk (see ). 2) The partial results of classification of simple Z 2 -graded Lie algebras were independently obtained by Kaplansky, Freund, Djokovic, Pais, Rittenberg, Nahm, Scheunert . Chapter I. Classification of Simple Lie Superalgebras § ί. Superalgebras and Lie Superalgebras 1. Superalgebras. Suppose A = AQ@AJ is a superalgebra. Call subalgebra or ideal of a superalgebra A a Z 2 -graded subalgebra or ideal. Call homomorphism the one that preserves Z 2 -grading. Direct and semidirect sum of superalgebras is defined as usual. The definition of tensor product is different. Suppose A = A 0 @A 1
We classify all linearly compact simple Jordan superalgebras over an algebraically closed field o... more We 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.
We complete the classification of conformal embeddings of a maximally reductive subalgebra k into... more We complete the classification of conformal embeddings of a maximally reductive subalgebra k into a simple Lie algebra g at non-integrable non-critical levels k by dealing with the case when k has rank less than that of g. We describe some remarkable instances of decomposition of the vertex algebra V k (g) as a module for the vertex subalgebra generated by k. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings A1 × A1 ֒→ C3 at level k = −1/2, and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding A1 × D4 ֒→ C8 at level k = −1/2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.
We prove that any unitary highest weight module over a universal minimal quantum affine W-algebra... more We prove that any unitary highest weight module over a universal minimal quantum affine W-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine W-algebras and the list of all their irreducible positive energy modules. We also classify all irreducible highest weight modules for the simple affine vertex algebras in the cases when the associated simple minimal W-algebra is unitary.
We prove that, for a Poisson vertex algebra V, the canonical injective homomorphism of the variat... more We prove that, for a Poisson vertex algebra V, the canonical injective homomorphism of the variational cohomology of V to its classical cohomology is an isomorphism, provided that V, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables. Contents 1. Introduction 1 2. Variational PVA cohomology 4 3. Preliminaries on the symmetric group and on graphs 6 4. Classical PVA cohomology 9 5. The Main Theorem 14 6. Sesquilinear Harrison cohomology 15 7. Relation between symmetric sesquilinear Harrison and classical PVA cohomology complexes 21 8. Vanishing of the sesquilinear Harrison cohomology 24 9. Proof of the Main Theorem 5.2 32 References 33 Key words and phrases. Poisson vertex algebra (PVA), classical operad, classical PVA cohomology, variational PVA cohomology, sesquilinear Hochschild and Harrison cohomology.
We introduce the formalism of differential conformal superalgebras, which we show leads to the "c... more We 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 superalgebras by means of Galois cohomology.
The problem of representing an integer as a sum of squares of integers has had a long history. On... more The problem of representing an integer as a sum of squares of integers has had a long history. One of the first after antiquity was A. Girard who in 1632 conjectured that an odd prime p can be represented as a sum of two squares iff p ≡ 1 mod 4, and P. Fermat in 1641 gave an "irrefutable proof" of this conjecture. The subsequent work on this problem culminated in papers by A.M. Legendre (1798) and C.F. Gauss (1801) who found explicit formulas for the number of representations of an integer as a sum of two squares. C.G. Bachet in 1621 conjectured that any positive integer can be represented as a sum of four squares of integers, and it took efforts of many mathematicians for about 150 years before J.-L. Lagrange gave a proof of this conjecture in 1770.
We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. ... more We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary W-algebra. We show that for a minimal simple W-algebra W k (g, θ/2) this form can be unitary only when its 1 2 Z-grading is compatible with parity, unless W k (g, θ/2) "collapses" to its affine subalgebra.
In this paper we study the complete reducibility of representations of infinitedimensional Lie al... more In this paper we study the complete reducibility of representations of infinitedimensional Lie algebras from the perspective of representation theory of vertex algebras.
Let g be a finite-dimensional semisimple Lie algebra and (• , •) its Killing form, σ an elliptic ... more Let g be a finite-dimensional semisimple Lie algebra and (• , •) its Killing form, σ an elliptic automorphism of g, and a a σ-invariant reductive subalgebra of g, such that the restriction of the form (• , •) to a is non-degenerate. Let L(g, σ) and L(a, σ) be the associated twisted affine Lie algebras and F σ (p) the σ-twisted Clifford module over L(a, σ), associated to the orthocomplement p of a in g. Under suitable hypotheses on σ and a, we provide a general formula for the decomposition of the kernel of the affine Dirac operator, acting on the tensor product of an integrable highest weight L(g, σ)-module and F σ (p), into irreducible L(a, σ)-submodules. As an application, we derive the decomposition of all level 1 integrable irreducible highest weight modules over orthogonal affine Lie algebras with respect to the affinization of the isotropy subalgebra of an arbitrary symmetric space.
We show that the normalized supercharacters of principal admissible modules over the affine Lie s... more We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra s 2|1 (resp. ps 2|2) can be modified, using Zwegers' real analytic corrections, to form a modular (resp. S-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N = 2 (resp. N = 4) superconformal algebras with central charge 3 1 − (2m + 2)/M , where m ∈ Z ≥0 , M ∈ Z ≥2 , gcd(2m + 2, M) = 1 if m > 0 (resp. 6 (m/M − 1), where m ∈ Z ≥1 , M ∈ Z ≥2 , gcd(2m, M) = 1 if m > 1), whose modified characters and supercharacters form a modular invariant family.
In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)al... more In 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 t his cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalg ebras.
We show that there are precisely two, up to conjugation, anti-involutions σ ± of the algebra of d... more We show that there are precisely two, up to conjugation, anti-involutions σ ± of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D ± of the Lie subalgebra of this algebra fixed by −σ ± , and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra C[u]/(u m+1) and its classical Lie subalgebras of B, C and D types. Character formulas for positive primitive representations of D ± (including all the unitary ones) are obtained. We also realize a class of primitive representations of D ± in terms of free fields and establish a number of duality results between these primitive representations and finite-dimensional irreducible representations of finite-dimensional Lie groups and supergroups. We show that the vacuum module V c of D + carries a vertex algebra structure and establish a relationship between V c for c ∈ 1 2 Z and W-algebras.
In the paper we consider the problem of computation of characters of relatively integrable irredu... more In 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 g. The problems consists of two parts. First, it is the reduction of the problem to the g-module F (L), where g is the associated to L integral Lie superalgebra and F (L) is an integrable irreducible highest weight gmodule. Second, it is the computation of characters of integrable highest weight modules. There is a general conjecture concerning the first part, which we check in many cases. As for the second part, we prove in many cases the KW-character formula, provided that the KW-condition holds, including almost all finite-dimensional g-modules when g is basic, and all maximally atypical non-critical integrable g-modules when g is affine with non-zero dual Coxeter number. Contents † Supported in part by BSF Grant No. 711623. ‡ Supported in part by Simons fellowship.
N ≤ 8 3-algebras have recently appeared in N-supersymmetric 3-dimensional Chern-Simons gauge theo... more N ≤ 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 ≥ 3. In the present paper we classify linearly compact simple N = 6 3-algebras, using their correspondence with simple linearly compact Lie superalgebras with a consistent short Z-grading, endowed with a graded conjugation. We also briefly discuss N = 5 3-algebras.
We complete the classification of conformal embeddings of a maximally reductive subalgebra k into... more We complete the classification of conformal embeddings of a maximally reductive subalgebra k into a simple Lie algebra g at non-integrable non-critical levels k by dealing with the case when k has rank less than that of g. We describe some remarkable instances of decomposition of the vertex algebra V k (g) as a module for the vertex subalgebra generated by k. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings A1 × A1 ֒→ C3 at level k = −1/2, and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding A1 × D4 ֒→ C8 at level k = −1/2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs.
International Journal of Geometric Methods in Modern Physics, Sep 1, 2006
We describe the group of continous automorphisms of all simple infinitedimensional linearly compa... more We describe the group of continous automorphisms of all simple infinitedimensional linearly compact Lie superalgebras and use it in order to classify F-forms of these superalgebras over any field F of characteristic zero.
We give a short introduction to generalized vertex algebras, using the notion of polylocal fields... more We 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 of rank equal to dim h and all irreducible representations of these vertex algebras.
Uploads
Papers by Victor Kac