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Victor G. Kac

Publications and source records attributed to Victor G. Kac.

At least 91 records · Page 5Linked to original sources
arXiv2011

Essential variational Poisson cohomology

In 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 functions is normal and is an algebra over a linearly closed differential field. In the present paper we show that, for K skewadjoint, this cohomology, viewed as a Z-graded Lie superalgebra, is isomorphic to the finite dimensional Lie superalgebra of Hamiltonian vector fields over a Grassman algebra. We also prove that the subalgebra of `essential' variational Poisson cohomology, consisting of classes vanishing on the Casimirs of K, is zero. This vanishing result has applications to the theory of bi-Hamiltonian structures and their deformations. At the end of the paper we consider also the translation invariant case.

math-ph↗
arXiv2010

Classification of linearly compact simple N=6 3-algebras

$N\leq 8$ 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 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.

math.QA↗
arXiv2010

Irreducible Modules over Finite Simple Lie Pseudoalgebras II. Primitive Pseudoalgebras of Type K

One 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 algebra for which C[\partial] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra. The finite (i.e., finitely generated over H) simple Lie pseudoalgebras were classified in our previous work. The present paper is the second in our series on representation theory of simple Lie pseudoalgebras. In the first paper we showed that any finite irreducible module over a simple Lie pseudoalgebra of type W or S is either an irreducible tensor module or the kernel of the differential in a member of the pseudo de Rham complex. In the present paper we establish a similar result for Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by Rumin.

math.QA↗
arXiv2010

Classification of simple linearly compact n-Lie superalgebras

We 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 L=\oplus_{j=-1}^{n-1} L_j, such that L_{-1}=g, where dim L_{n-1}=1, L_{-1} and L_{n-1} generate L, and [L_j, L_{n-j-1}] =0 for all j, thereby reducing it to the known classification of simple linearly compact Lie superalgebras and their Z-gradings. The list consists of four examples, one of them being the n+1-dimensional vector product n-Lie algebra, and the remaining three infinite-dimensional n-Lie algebras.

math.QA↗
arXiv2009

Poisson vertex algebras in the theory of Hamiltonian equations

We 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 Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms ω_j, j in Z_+, of the variational complex Ω. If these forms are exact, i.e. ω_j are variational derivatives of some local functionals \int h_j, then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Ωis exact, provided that the algebra of functions V is "normal"; in particular, for arbitrary V, any closed form in Ωbecomes exact if we add to V a finite number of antiderivatives. We demonstrate on the examples of KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.

math-ph↗
arXiv2009

Classification of linearly compact simple rigid superalgebras

The 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 an isomorphic algebra. In this paper we classify all linearly compact simple anti-commutative (resp. commutative) rigid superalgebras. Beyond Lie (resp. Jordan) superalgebras the complete list includes four series and twenty two exceptional superalgebras (resp. ten exceptional superalgebras).

math.QA↗
arXiv2009

On the Kernel of the affine Dirac operator

Let L be a finite-dimensional semisimple Lie algebra with a non-degenerate invariant bilinear form, σan elliptic automorphism of L leaving the form invariant, and A a σ-invariant reductive subalgebra of L, such that the restriction of the form to A is non-degenerate. Consider the associated twisted affine Lie algebras L^, A^, and let F be the σ-twisted Clifford module over A^ associated to the orthocomplement of A in L. Under suitable hypotheses onσand 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^-module and F, into irreducible A^-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.

math.RT↗
arXiv2008

Simple Jordan conformal superalgebras

We classify simple finite Jordan conformal superalgebras and establish preliminary results for the classification of simple finite Jordan pseudoalgebras.

math.QA↗
arXiv2007

On rationality of W-algebras

We 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.

math-ph↗
arXiv2007

SUSY Lattice Vertex Algebras

We 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 form $(a,b) = 2ab$.

math.QA↗
arXiv2007

Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting

We 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 an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology. For our calculations we use the machinery of Lie conformal and vertex algebras.

math.RT↗
arXiv2006

Supersymmetric vertex algebras

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

math.QA↗
arXiv2006

Generalized Vertex Algebras

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.

math.QA↗