Relating Kac-Moody,Virasoro and Krichever-Novikov algebras

We demonstrate that the Kac-Moody and Virasoro-like algebras on Riemann surfaces of arbitrary genus with two punctures introduced by Krichever and Novikov are in two ways linearly related to Kac-Moody and Virasoro algebras onS ¹. The two relations differ by a Bogoliubov transformation, and we discuss the connection with the operator formalism.     

Virasoro type Lie algebras and deformations

The first and second cohomologies of Cartan Type Lie algebras with coefficients in irreducible tensor modules are calculated. The spaceH ¹(L, U) is interpreted as a space of deformations of (L, U)-modules.H ²(L, L)≠0 ifL=S ₂,S ₂ ⁺ orL=H _n ,H _n ⁺ . Lie algebra of divergenceless vector fieldsS ₂ ⁺ has only one nontrivial local deformation. The two-sided simple hamiltonian algebraH _n has 2n ²+n new local deformations in addition to Moyal cocycle. The Lie algebrasL=W _n (n>3),S _n?1(n>2),H _n (n>1),K _n+1(n>1) have 3, 1, 1, 3 nonisomorphic tensor modules with irreducible bases and nonzero 1-cohomologies; respectively, the corresponding numbers for 2-cohomologies are 9, 6, 7 and 9.     

Higher-Dimensional Generalizations of Affine Kac-Moody and Virasoro Conformal Lie Algebras

We discuss the generalizations of the notion of Conformal Algebra and Local Distribution Lie algebras for multi-dimensional bases. We replace the algebra of Laurent polynomials on by an infinite-dimensional representation (with some additional structures) of a simple finite-dimensional Lie algebra in the space of regular functions on the corresponding Grassmann variety that can be described as a ``right' higher-dimensional generalization of from the point of view of a corresponding group action. For it gives us the usual Vertex Algebra notion. We construct the higher dimensional generalizations of the Virasoro and the Affine Kac-Moody Conformal Lie algebras explicitly and in terms of the Operator Product Expansion.     

Step algebras of semi-simple subalgebras of Lie algebras

For each pair (G,K) where G is a complex finite-dimensional Lie algebra and K a semi-simple subalgebra of G, we construct an associative algebra (step algebra) $\text{Y}$ (G,K) and a homomorphism i_*: $\text{Y}$ (G,K)→E(G) is the enveloping algebra of G. $\text{Y}$ (G,K) has the following properties: (1) If V is any G-module and x ? V a K-maximal vector, then sx = i_* (s)x is K-maximal for any s ? $\text{Y}$ (G,K); (2) If V is irreducible and a certain simple criteria is fulfilled, then any K-maximal vector can be written in the form sx_m, s ? $\text{Y}$ (G,K), where x_m is some fixed K-maximal vector. Because of these properties $\text{Y}$ (G,K) has great practical value when constructing irreducible representations of Lie algebras in a form which makes the reduction with respect to a semi-simple subalgebra explicit.     

Lie algebras associated with topological nilpotent groups

A Lie algebra structure is defined on the set of all continuous one-parameter groups of nilpotent topological groups. Extensions are given to some inductive and projective limits.     

Vertex representations forN-toroidal Lie algebras and a generalization of the Virasoro algebra

Vertex representations are obtained for toroidal Lie algebras for any number of variables. These representations afford representations of certainn-variable generalizations of the Virasoro algebra that are abelian extensions of the Lie algebra of vector fields on a torus.Work supported in part by the Natural Sciences and Engineering Research Council of Canada.     

On the isotropy subalgebras of Lie algebras of conformal vector fields

The conformal isotropy algebra of a point m in an n-manifold with a metric of arbitrary signature is shown to be locally reducible, by a conformal change of the metric, to a homothetic algebra near m iff, by choice of a chart, its constituent vector fields are simultaneously linearisable at m and, for n≥3, a necessary and sufficient condition for this in terms of the first and second derivatives of these fields at m is given. The implications for the Riemannian case and the Lorentzian case are investigated. In contrast to the former, a Lorentzian manifold admitting a conformal vector field that is not linearisable at some point need not be conformally flat. Relevant four-dimensional examples are provided.     

Semi-simple real subalgebras of non-compact semi-simple real Lie algebras IV

The investigation of the problem of embedding a semi-simple real Lie algebra $\text{L}$′ in a non-compact semi-simple real Lie algebra $\text{L}$ is extended to the case in which at least one of the real Lie algebras has a semi-simple complex extension, which consists of the direct sum of two simple complex Lie algebras. Detailed procedures are given, which together with those given previously, allow the construction of all embeddings of $\text{L}$′ in $\text{L}$ when their complex extensions are A₁, B₁, C₁, D₁ or a direct sum of any two of these. The procedures are illustrated by considering examples corresponding to complex Lie algebra embeddings A₁?(A₂⊕A₂), (A₁⊕A₁)?(A₂⊕A₂), (A₁⊕A₁)?A₃, (A₁⊕A₁)??(A₃⊕A₃) and (A₁⊕A₁)?(A₃⊕A₂). Because of its physical significanc embeddings of SL(2,C) in simple and semi-simple real Lie algebras are studied in detail.     

Analytic vectors and irreducible representations of nilpotent Lie groups and algebras

Let U be a unitary irreducible locally faithful representation of a nilpotent Lie group G, U the universal enveloping algebra of G, M a simple module on U with kernel Ker dU, then there exists an automorphism of U keeping ker dU invariant such that, after transport of structure, M is isomorphic to a submodule of the space of analytic vectors for U.     

Pseudo-Kähler metrics on six-dimensional nilpotent Lie algebras

It is well known that any symplectic manifold (M,Ω) has an almost complex structure J which is compatible with Ω. In this paper, we deal with the existence of compatible pairs (J,Ω) on nilpotent Lie algebras of dimension ≤6, J being an integrable almost complex structure. We prove that if such a pair exists, J must satisfy some extra conditions, namely J must be nilpotent in the sense of [Trans. Am. Math. Soc. 352 (2000) 5405]. Associated to any such a compatible pair, there is a pseudo-Kähler metric g which cannot be positive definite unless be abelian. All these metrics are Ricci flat, although many of them are nonflat, and we study the behaviour of its curvature tensor under deformation.     

Supersymmetry and Kac-Moody algebras

The invariance algebra of the Majorana action contains a Kac-Moody algebra which, on shell, reduces to an Abelian algebra. In the absence of auxiliary fields in the Wess-Zumino model, supersymmetry transformations generate an infinite-dimensional Lie algebra, which is shown to be a Grassmannian extension of this Kac-Moody algebra. The corresponding Noether charges are discussed.     

Determinant bundles and Virasoro algebras

We consider the interplay of infinite-dimensional Lie algebras of Virasoro type and moduli spaces of curves, suggested by string theory. We will see that the infinitesimal geometry of determinant bundles is governed by Virasoro symmetries. The Mumford forms are just invariants of these symmetries. The representations of Virasoro algebra define (twisted)D-modules on moduli spaces; theseD-modules are equations on correlators in conformal field theory.To the memory of Vadik Knizhnik (20. 2. 1962–25. 12. 1987)     

Kac-Moody group representations and generalization of the Sugawara construction of the Virasoro algebra

We discuss the dynamical structure of the semidirect product of the Virasoro and affine Kac-Moody groups within the framework of a group quantization formalism. This formalism provides a realization of the Virasoro algebra acting on Kac-Moody Fock states which generalizes the Sugawara construction. We also give an explicit construction of the standard Kac-Moody group representations associated with strings on SU(2) and recover, in particular, the renormalization factor of L(z) Research partially supported by the Conselleria de Cultura de la Generalitat Valenciana, the Plan de Formacion del Personal Investigador, the Comision Asesora de Investigacion Cientifica y Tecnica (CAICYT), and The British Council.     

Affine Kac-Moody algebras and semi-infinite flag manifolds

We study representations of affine Kac-Moody algebras from a geometric point of view. It is shown that Wakimoto modules introduced in [18], which are important in conformal field theory, correspond to certain sheaves on a semi-infinite flag manifold with support on its Schhubert cells. This manifold is equipped with a remarkable semi-infinite structure, which is discussed; in particular, the semi-infinite homology of this manifold is computed. The Cousin-Grothendieck resolution of an invertible sheaf on a semi-infinite flag manifold gives a two-sided resolution of an irreducible representation of an affine algebras, consisting of Wakimoto modules. This is just the BRST complex. As a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent Lie algebra.Dedicated to Dmitry Borisovich Fuchs on his 50th birthdayAddress after September 15, 1989: Mathematics Department, Harvard University, Cambrdige, MA 02138, USA     

Unitary representations of the Virasoro and super-Virasoro algebras

It is shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine Kac-Moody algebras yields the full discrete series of highest weight irreducible representations of the Virasoro algebra. The corresponding method for the super-Virasoro algebras (i.e. the Neveu-Schwarz and Ramond algebras) is described in detail and shown to yield the full discrete series of irreducible highest weight representations.     

Virasoro algebras in the theory of bosonicp-branes

It is shown that in the case of closed bosonicp-branes there existp + 1 mutually commuting Virasoro algebras which are a direct generalization of the string case. The existence of these algebras allows us to conclude that a non-Abelian string spectrum and quantum anomalies are admissible. The generalization of these results for the supersymmetric case is also discussed.     

Vertex operator representation of some quantum tori Lie algebras

We are defining the trigonometric Lie subalgebras in which are the natural generalization of the well known Sin-Lie algebra. The embedding formulas into are introduced. These algebras can be considered as some Lie algebras of quantum tori. An irreducible representation ofA, B series of trigonometric Lie algebras is constructed. Special cases of the trigonometric Lie factor algebras, which can be considered as a quantum (preserving Lie algebra structure) deformation of the Kac-Moody algebras are considered.     

Cohomology of the Virasoro algebra with coefficients in string fields

The first cohomology of the Virasoro algebra with coefficients in string fields are investigated. The relation between them and the Nambu-Goto action for a closed string is established.     

Generalised bosonic construction of Virasoro algebras

Energy-momentum tensors of conformal field theories and some of their primary fields, including those of parafermionic theories based on simply-laced Lie algebras, are constructed from free bosons. The classification of such theories requires a generalisation of the root systems of Lie algebras. The complete list of such energy-momentum tensors, that can be constructed from two free bosons, includes those of the first four c<1 theories.