The Lie algebra of the sl(2, ℂ)-valued automorphic functions on a torus

It is shown that the Lie algebra of the automorphic, meromorphic sl(2, )-valued functions on a torus is a geometric realization of a certain infinite-dimensional finitely generated Lie algebra. In the trigonometric limit, when the modular parameter of the torus goes to zero, the former Lie algebra goes over into the sl(2, )-valued loop algebra, while the latter goes into the Lie algebra (A ₁ ⁽¹⁾ )/(centre).     

An existence result for super Lie groups

An analogue of the classical global third Lie theorem is proved to hold for super Lie groups whose ground Banach—Grassmann algebra is (possibly) infinite-dimensional, provided that this algebra has an ordered basis. It is also proved that the superanalytic structure of a connected super Lie group having a prescribed Lie module is unique, although in a weaker sense than in the case of ordinary Lie groups.Work partly supported by the National Group for Mathematical Physics (GNFM) of the Italian National Research Council (CNR), and by the Italian Ministry of Education through the research project Geometry and Physics.     

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.     

Continuum analogues of contragredient Lie algebras (Lie algebras with a Cartan operator and nonlinear dynamical systems)

We present an axiomatic formulation of a new class of infinitedimensional Lie algebras-the generalizations ofZ-graded Lie algebras with, generally speaking, an infinite-dimensional Cartan subalgebra and a contiguous set of roots. We call such algebras continuum Lie algebras. The simple Lie algebras of constant growth are encapsulated in our formulation. We pay particular attention to the case when the local algebra is parametrized by a commutative algebra while the Cartan operator (the generalization of the Cartan matrix) is a linear operator. Special examples of these algebras are the Kac-Moody algebras, algebras of Poisson brackets, algebras of vector fields on a manifold, current algebras, and algebras with differential or integro-differential cartan operator. The nonlinear dynamical systems associated with the continuum contragredient Lie algebras are also considered.     

Some Infinite-Dimensional Algebras Arising in Spin Systems and in Particle Physics and their Grand Algebra

We indicate similarities in the structure of two types of infinite-dimensional algebras, one introduced 28 years ago in connection with the mass problem of elementary particles and the other seven years ago in connection with spin systems (XY models). We show that these algebras can be considered as representations of a single Grand Algebra, the enveloping algebra of an affine Kac–Moody algebra built on the Poincaré Lie algebra. As an associative and coassociative bialgebra of operators, the latter representation of the grand algebra is a preferred nontrivial deformation of the Ising case bialgebra.     

A supersymmetric extension of the Toda lattice hierarchy

A supersymmetric extension of the Toda lattice (STL) hierarchy is introduced. Explicit representation of solutions of the STL hierarchy is given by means of the Riemann-Hilbert decomposition. The STL hierarchy connected with the infinite-dimensional Lie super algebra osp (/) is studied.     

Classification of all Poisson-Lie structures on an infinite-dimensional jet group

A local classification of all Poisson-Lie structures on an infinite-dimensional group G of formal power series is given. All Lie bialgebra structures on the Lie algebra G of G are also classified.     

Generalizing the $$\mathfrak {bms}_{3}$$ and 2D-conformal algebras by expanding the Virasoro algebra

By means of the Lie algebra expansion method, the centrally extended conformal algebra in two dimensions and the \(\mathfrak {bms}_{3}\) algebra are obtained from the Virasoro algebra. We extend this result to construct new families of expanded Virasoro algebras that turn out to be infinite-dimensional lifts of the so-called \(\mathfrak {B}_{k}\), \(\mathfrak {C}_{k}\) and \(\mathfrak {D}_{k}\) algebras recently introduced in the literature in the context of (super)gravity. We also show how some of these new infinite-dimensional symmetries can be obtained from expanded Ka?–Moody algebras using modified Sugawara constructions. Applications in the context of three-dimensional gravity are briefly discussed.     

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.     

Some remarks on Lie superalgebras

We propose an approach to the theory of Lie superalgebras based on what we call a Lie algebra square root. Every Lie algebra square root has a Lie algebra as its square, but many different Lie algebra square roots may have the same square.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

On \mathbb{Z}-Gradations of Twisted Loop Lie Algebras of Complex Simple Lie Algebras

We define the twisted loop Lie algebra of a finite dimensional Lie algebra as the Fréchet space of all twisted periodic smooth mappings from to . Here the Lie algebra operation is continuous. We call such Lie algebras Fréchet Lie algebras. We introduce the notion of an integrable -gradation of a Fréchet Lie algebra, and find all inequivalent integrable -gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.On leave of absence from the Institute for Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia.     

Usage of infinite-dimensional nuclear algebras in superanalysis

An infinite-dimensional topological algebra is defined as an inductive limit of finite-dimensional -commutative Banach algebras. This algebra has some desirable properties for the algebra of supernumbers, on which we can develop a satisfactory theory of superanalysis.     

Quantum integrable systems and oscillator representations of c n

Solutions of the Yang-Baxter equation with spectral parameter for systems with in-variance under a Lie algebra and for which the quantum space is a Hilbert space different from the auxiliary space are studied. In particular, for the case of =c_n= sp (2n, ), solutions on infinite-dimensional state spaces are constructed.     

On a class of local Lie algebras over a manifold

This letter presents a study of the automorphisms and the derivations of a large class of local Lie algebras over a manifold M (in the sense of Shiga and Kirillov) called Lie algebras of order O over M.It is shown that, in general, the algebraic structure of such an algebra characterizes the differentiable structure of M and that the Lie algebra of derivations of is a Lie algebra of differential operators of order 1 over M obtained in a natural way as the space of sections of a vector bundle canonically associated to .     

Yang–Mills Algebra

Some unexpected properties of the cubic algebra generated by the covariant derivatives of a generic Yang–Mills connection over the (s+1)-dimensional pseudo Euclidean space are pointed out. This algebra is Koszul of global dimension 3 and Gorenstein but except for s=1 (i.e. in the two-dimensional case) where it is the universal enveloping algebra of the Heisenberg Lie algebra and is a cubic Artin–Schelter regular algebra, it fails to be regular in that it has exponential growth. We give an explicit formula for the Poincaré series of this algebra and for the dimension in degree n of the graded Lie algebra of which is the universal enveloping algebra. In the four-dimensional (i.e. s=3) Euclidean case, a quotient of this algebra is the quadratic algebra generated by the covariant derivatives of a generic (anti) self-dual connection. This latter algebra is Koszul of global dimension 2 but is not Gorenstein and has exponential growth. It is the universal enveloping algebra of the graded Lie algebra which is the semi-direct product of the free Lie algebra with three generators of degree one by a derivation of degree one.     

A Super-Analogue of Kontsevich’s Theorem on Graph Homology

In this Letter, we will prove a super-analogue of a well-known result by Kontsevich which states that the homology of a certain complex generated by isomorphism classes of oriented graphs can be calculated as the Lie algebra homology of an infinite-dimensional Lie algebra of symplectic vector fields.     

Universal Characters and an Extension of the KP Hierarchy

The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we define vertex operators which play roles of raising operators for the universal character. By means of the vertex operators, we obtain a series of non-linear partial differential equations of infinite order, called the UC hierarchy; we regard it as an extension of the KP hierarchy. We investigate also solutions of the UC hierarchy; the totality of the space of solutions forms a direct product of two infinite-dimensional Grassmann manifolds, and its infinitesimal transformations are described in terms of the Lie algebra .     

Extended structures and new infinite-dimensional hidden symmetries for the Einstein–Maxwell-dilaton-axion theory with multiple vector fields

A so-called extended elliptical-complex (EEC) function method is proposed and used to further study the Einstein–Maxwell-dilaton-axion theory with p vector fields (EMDA-p theory, for brevity) for . An Ernst-like matrix EEC potential is introduced and the motion equations of the stationary axisymmetric EMDA-p theory are written as a so-called Hauser–Ernst-like self-dual relation for the EEC matrix potential. In particular, for the EMDA-2 theory, two Hauser–Ernst-type EEC linear systems are established and based on their solutions some new parametrized symmetry transformations are explicitly constructed. These hidden symmetries are verified to constitute an infinite-dimensional Lie algebra, which is the semidirect product of the Kac–Moody algebra and Virasoro algebra (without centre charges). These results show that the studied EMDA-p theories possess very rich symmetry structures and the EEC function method is necessary and effective.     

Poincaré partially integrable local representations and mass-spectrum

An example is given of an irreducible representation of a finite-dimensional Lie algebra containing the Poincaré Lie algebra and giving rise to isolated positive masses. In addition the representation is Poincaré partially integrable (which assures the continuous physical spectrum for the energy- momentum vector) and Poincaré-covariant in a weak sense.A connection between this example and some recently published impossibility theorems is shown, and conclusions about a possible future work in this domain are also drawn.