Central Extensions Of Lie Algebras Research Articles

Five-dimensional p-nilpotent restricted Lie algebras over algebraically closed fields of characteristic p > 3

A classification of p-nilpotent 5-dimensional restricted Lie algebras over algebraically closed fields of characteristic p>3 is provided. This is achieved by employing a natural restricted analogue of the known method by Skjelbred and Sund for classifying ordinary nilpotent Lie algebras as central extensions of Lie algebras of smaller dimension.

On S-expansions and other transformations of Lie algebras

The aim of this work is to study the relation between S-expansions and other transformations of Lie algebras. In particular, we prove that contractions, deformations and central extensions of Lie algebras are preserved by S-expansions. We also provide several examples and give conditions so transformations of reduced subalgebras of S-expanded algebras are preserved by the S-expansion procedure.

Automorphisms, derivations, and central extensions of Lie algebras arising from quantum tori

For a Lie algebra related to a quantum torus, we compute its automorphisms, derivations, and universal central extension. This Lie algebra is isomorphic to a subalgebra of the Lie algebra of derivations over the quantum torus, and moreover contains a subalgebra isomorphic to the centerless higher rank Virasoro algebra.

Homotopy moment maps

Associated to any manifold equipped with a closed form of degree >1 is an ‘L∞-algebra of observables’ which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L∞-morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L∞-algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called ‘string Lie 2-algebra’ arises this way.

Integrating central extensions of Lie algebras via Lie 2-groups

The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of \[Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of $\pi\_2$ for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial $\pi\_2$2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of étale Lie 2-groups. As an application, we obtain a generalization of Lie’s Third Theorem to infinite-dimensional Lie algebras.

Central extensions of Lie algebras of symplectic and divergence free vector fields

In this review paper, we present several results on central extensions of the Lie algebra of symplectic (Hamiltonian) vector fields, and compare them with similar results for the Lie algebra of (exact) divergence free vector fields. In particular, we co

Central Extensions of Root Graded Lie Algebras

We study the central extensions of Lie algebras graded by an irreducible locally finite root system.

DJKM algebras and non-classical orthogonal polynomials

We describe families of polynomials arising in the study of the universal central extensions of Lie algebras introduced by Date, Jimbo, Kashiwara, and Miwa (1983) [6] in their work on the Landau–Lifshitz equations. We show these two families of polynomials satisfy certain fourth order linear differential equations by direct computation and one of the families is a particular collection of associated ultraspherical polynomials.

Isoclinic extensions of Lie algebras

In this article we introduce the notion of the equivalence relation, isoclinism, on the central extensions of Lie algebras, and determine all central extensions occurring in an isoclinism class of a given central extension. We also show that under some conditions, the concepts of isoclinism and isomorphism between the central extensions of finite dimensional Lie algebras are identical. Finally, the connection between isoclinic extensions and the Schur multiplier of Lie algebras are discussed.

ON UNIVERSAL COVERINGS OF LIE TORI

Abstract. In this paper we give an introduction to the theory of uni-versal central extensions of perfect Lie algebras. In particular, we willprovide a model for the universal coverings of Lie tori and we show thatautomorphisms and derivations lift to the universal coverings. We alsoprove that the universal covering of a Lie Λ-torus of type ∆ is again a LieΛ-torus of type ∆. 0. IntroductionCentral extensions play an important role in the theory of Lie algebras.Universal central extension of Lie algebras over rings were described in [23, §1],later references on universal central extensions are [12, §1], [15, 1.9], [24, 7.9] or[21]. Central extensions in the category of certain topological Lie algebras arestudied in [17]. Garland studies universal central extensions of Lie algebrasoverfields [12, §1]. In particular, he constructsa model ofuniversalcentralextensionof a perfect Lie algebra, using the universal 2-cocycle, which is different fromthe Van der Kallen , s model (see [23, §1]). Our construction for universal centralextensions of Lie tori is essentially the Van der Kallen’s model.Lie tori play a critical role in the theory of extended affine Lie algebras whichare natural generalization of finite dimensional simple Lie algebras and affineLie algebras. Yoshii and Neher were interested in Lie tori primarily becauseof the connection between Lie tori and extended affine Lie algebras ([26] and[19]). The centerless core of an extended affine Lie algebra is a centerless ofLie torus and conversely any centerless Lie torus is the centerless core of anextended affine Lie algebra [26, Theorem 7.3]. Extended affine Lie algebrasare defined axiomatically by Alison, Azam, Berman, Gao and Pianzola in [1].The various classes of these Lie algebras have been investigated in many papers(see [4, 6, 13, 14, 20, 25, 27] and [16]). Lie tori as well as extended affine Lie

Versal deformations and versality in central extensions of Jacobi schemes

Let \(\mathfrak{L}_m \) be the scheme of the laws defined by the Jacobi identities on \(\mathbb{K}^m \) with \(\mathbb{K}\) a field. A deformation of \(\mathfrak{g} \in \mathfrak{L}_m \), parametrized by a local ring A, is a local morphism from the local ring of \(\mathfrak{L}_m \) at ϕm to A. The problem of classifying all the deformation equivalence classes of a Lie algebra with given base is solved by “versal” deformations. First, we give an algorithm for computing versal deformations. Second, we prove there is a bijection between the deformation equivalence classes of an algebraic Lie algebra ϕm = R ⋉ φn in \(\mathfrak{L}_m \) and its nilpotent radical φn in the R-invariant scheme \(\mathfrak{L}_{n}^{\rm R} \) with reductive part R, under some conditions. So the versal deformations of ϕm in \(\mathfrak{L}_m \) are deduced from those of φn in \(\mathfrak{L}_{n}^{\rm R} \), which is a more simple problem. Third, we study versality in central extensions of Lie algebras. Finally, we calculate versal deformations of some Lie algebras.

A cohomological characterization of Leibniz central extensions of Lie algebras

Mainly motivated by Pirashvili's spectral sequences on a Leibniz algebra, a cohomological characterization of Leibniz central extensions of Lie algebras is given. In particular, as applications, we obtain the cohomological version of Gao's main theorem for Kac-Moody algebras and answer a question in an earlier paper by Liu and Hu (2004).

The restricted Grassmannian, Banach Lie–Poisson spaces, and coadjoint orbits

We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie–Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie–Poisson spaces, and the underlying Banach space can be chosen to be even a Hilbert space. Smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established. Several pathological properties of the restricted algebra are pointed out.

Twisted modules for vertex operator algebras and Bernoulli polynomials

This work is a continuation of a series of papers of two of the present authors [L3], [L4], [M1]–[M3], stimulated by work of Bloch [Bl]. In those papers we used the general theory of vertex operator algebras to study central extensions of classical Lie algebras and superalgebras of differential operators on the circle in connection with values of ζ–functions at the negative integers. In the present paper, using general principles of the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. This is explained through results in the general theory of vertex operator algebras, including a new identity, which we call “modified weak associativity.” This paper is an announcement. The detailed proofs will appear elsewhere. More specifically, the present goal is to obtain a new general Jacobi identity for twisted operators, and for related iterates of such operators, extending the previous analogous results in the untwisted setting in our papers mentioned above. As a consequence we obtain twisted constructions of certain central extensions of Lie algebras of differential operators on the circle, combining and extending methods from [L3], [L4], [M1]–[M3], [FLM1], [FLM2] and [DL]. In those earlier papers we used vertex operator techniques to analyze untwisted actions of the Lie algebra D, studied in [Bl], on a module for a Heisenberg Lie algebra of a certain standard type, based on a finite-dimensional vector space equipped with a nondegenerate symmetric bilinear form. Now consider an arbitrary isometry ν of period say p, that is, with ν = 1. Here we announce that the corresponding ν–twisted modules carry an action of the Lie algebra D in terms of twisted vertex operators, parametrized by certain quadratic vectors in the untwisted module. In particular, we extend a result from [FLM1], [FLM2], [DL] where actions of the Virasoro algebra were constructed using twisted vertex operators. In addition we explicitly compute certain “correction” terms for the generators of the “Cartan subalgebra” of D that naturally appear in any twisted construction. These correction terms are expressed in terms of special values of certain Bernoulli polynomials. They can in principle be generated, in the theory of vertex operator algebras, by the formal operator ex [FLM1], B.D. gratefully acknowledges partial support from an NSERC Postgraduate Scholarship. J.L. and A.M. gratefully acknowledge partial support from NSF grant DMS-0070800.

Central extensions of Lie algebras graded by finite root systems

Lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by Berman and Moody, Benkart and Zelmanov, and Neher. In this paper we determine the central extensions of these Lie algebras and hence describe them completely up to isomorphism.

Cohomology of Lie superalgebras and their generalizations

The cohomology groups of Lie superalgebras and, more generally, of ε Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is nontrivial. Two general propositions are proved, which help to calculate the cohomology groups. Several examples are included to show the peculiarities of the super case. For L=sl(1|2), the cohomology groups H1(L,V) and H2(L,V), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H2(L,U(L)) [with U(L) the enveloping algebra of L] is trivial. This implies that the superalgebra U(L) does not admit any nontrivial formal deformations (in the sense of Gerstenhaber). Garland’s theory of universal central extensions of Lie algebras is generalized to the case of ε Lie algebras.

Extensions centrales d'algèbres et de groupes de lie de dimension infinie, algèbre de virasoro et généralisations

This article surveys problems related to central extensions of Lie algebra of vector fields, both from the pure algebraic point of view (cohomological computations) and from the point of view of the geometrical and physical applications. Especially the cases of hamiltonian, contact and unimodular vector fields are developed, including the applications to fluid mechanics.

Generalized quantization scheme for central extensions of Lie algebras

We present the method for finding of the nonlinear Poisson-Lie groups structures on the vector spaces and for their quantization. For arbitrary central extension of Lie algebra explicit formulas of quantization are proposed.

Non-trivial central extensions of Lie algebras of differential operators in two and higher dimensions

Explicit formulas for the central extensions for Lie algebras of differential and pseudodifferential operators on the supercircle S1∣N and the general n-dimensional manifold Mn are given. Possible applications of these formulas to gravity theories coupled with higher spin fields in two and higher dimensions are discussed.