Infinite-dimensional Superalgebras Research Articles

On the supersymmetric extension of asymptotic symmetries in three spacetime dimensions

In this work we obtain known and new supersymmetric extensions of diverse asymptotic symmetries defined in three spacetime dimensions by considering the semigroup expansion method. The super-BMS_3, the superconformal algebra and new infinite-dimensional superalgebras are obtained by expanding the super-Virasoro algebra. The new superalgebras obtained are supersymmetric extensions of the asymptotic algebras of the Maxwell and the mathfrak {so}(2,2)oplus mathfrak {so}(2,1) gravity theories. We extend our results to the mathcal {N}=2 and mathcal {N}=4 cases and find that R-symmetry generators are required. We also show that the new infinite-dimensional structures are related through a flat limit ell rightarrow infty .

Brackets, superalgebras and spectral gap

The purpose of this survey is to discuss Poisson and contact brackets and related infinite dimensional superalgebras. All vector spaces are considered over the field of complex numbers $${\mathbb {C}}$$ .

Hom-Gel’fand–Dorfman super-bialgebras and Hom-Lie conformal superalgebras

The aim of this paper is to introduce and study Hom-Gel’fand–Dorfman super-bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing Hom-Gel’fand–Dorfman super-bialgebras. Also, we obtain some infinite-dimensional Hom-Lie superalgebras from affinization of Hom-Gel’fand–Dorfman super-bialgebras. Finally, we give a general construction of Hom-Lie conformal superalgebras from Hom-Lie superalgebras and establish the equivalence between quadratic Hom-Lie conformal superalgebras and Hom-Gel’fand–Dorfman super-bialgebras.

Natural Filtrations of Infinite-Dimensional Modular Contact Superalgebras

The natural filtration of the infinite-dimensional contact superalgebra over an algebraic closed field of positive characteristic is proved to be invariant under automorphisms by characterizing ad-nilpotent elements and the subalgebras generated by certain ad-nilpotent elements. Moreover, we obtain an intrinsic characterization of contact superalgebras and a property of automorphisms of these Lie superalgebras.

Hidden symmetries and killing tensors on curved spaces

Higher-order symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing-Yano tensors and nonstandard supersymmetries is pointed out. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac-type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The general results are applied to space-times which appear in modern studies. One presents the infinite dimensional superalgebra of Dirac type operators on the 4-dimensional Euclidean Taub-NUT space that can be seen as a twisted loop algebra. The existence of the conformal Killing-Yano tensors is investigated for some spaces with mixed 3-Sasakian structures.

Non-standard symmetries and Killing tensors

Higher order symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing-Yano tensors and non-standard supersymmetries is pointed out. The gravitational anomalies are absent if the hidden symmetry is associated with a Killing-Yano tensor. In the Dirac theory on curved spaces, Killing-Yano tensors generate Dirac type operators involved in interesting algebraic structures as dynamical algebras or even infinite dimensional algebras or superalgebras. The general results are applied to space-times which appear in modern studies. The 4-dimensional Euclidean Taub-NUT space and its generalizations introduced by Iwai and Katayama are analyzed from the point of view of hidden symmetries. One presents the infinite dimensional superalgebra of Dirac type operators on Taub-NUT space that can be seen as a twisted loop algebra. The axial anomaly, interpreted as the index of the Dirac operator, is computed for the generalized Taub-NUT metrics. The existence of the conformal Killing-Yano tensors is investigated for some spaces with mixed Sasakian structures.

Nonlinear symmetries on spaces admitting Killing tensors

Nonlinear symmetries corresponding to Killing tensors are investigated. The intimate relation between Killing–Yano tensors and non-standard supersymmetries is pointed out. The gravitational anomalies are absent if the hidden symmetry is associated with a Killing–Yano tensor. In the case of the nonlinear symmetries the dynamical algebras of the Dirac-type operators is more involved and could be organized as infinite dimensional algebras or superalgebras. The general results are applied to some concrete spaces involved in theories of modern physics. As a first example it is considered the 4-dimensional Euclidean Taub-NUT space and its generalizations introduced by Iwai and Katayama. One presents the infinite dimensional superalgebra of Dirac type operators on Taub-NUT space that could be seen as a graded loop superalgebra of the Kac-Moody type. The axial anomaly, interpreted as the index of the Dirac operator, is computed for the generalized Taub-NUT metrics. Finally the existence of the conformal Killing–Yano tensors is investigated for some spaces with mixed Sasakian structures.

Infinite loop superalgebras of the Dirac theory on the Euclidean Taub–NUT space

The Dirac theory in the Euclidean Taub–NUT space gives rise to a large collection of conserved operators associated with genuine or hidden symmetries. They are involved in interesting algebraic structures as dynamical algebras or even infinite-dimensional algebras or superalgebras. One presents here the infinite-dimensional superalgebra specific to the Dirac theory in manifolds carrying the Gross–Perry–Sorkin monopole. It is shown that there exists an infinite-dimensional superalgebra that can be seen as a twisted loop superalgebra.

Infinite Dimensional Lie and Jordan Algebras

The aim of this work is to provide a survey of some of the main structural results about graded algebras, in both, Lie and Jordan cases and relate them with some results about infinite dimensional superalgebras.

Simple Special Jordan Superalgebras with Associative Nil-Semisimple Even Part

We describe unital simple special Jordan superalgebras with associative nil-semisimple even part. In every such superalgebra J=A+M, either M is an associative and commutative A-module, or the associator space (A,A,M) coincides with M. In the former case, if JJ is not a superalgebra of the non-degenerate bilinear superform then its even part A is a differentiably simple algebra and its odd part M is a finitely generated projective A-module of rank 1. Multiplication in M is defined by fixed finite sets of derivations and elements of A. If, in addition, M is one-generated then the initial superalgebra is a twisted superalgebra of vector type. The condition of being one-generated for M is satisfied, for instance, if A is local or isomorphic to a polynomial algebra. We also give a description of superalgebras for which (A,A,M)≠0 and M⋂[A,M]≠0, where [,] is a commutator in the associative enveloping superalgebra of J. It is shown that such each infinite-dimensional superalgebra may be obtained from a simple Jordan superalgebra whose odd part is an associative module over the even.

Hilbert norms for graded algebras

Abstract: This paper presents a solution to a problem from superanalysis about the existence of Hilbert-Banach superalgebras. Two main results are derived: 1) There exist Hilbert norms on some graded algebras (infinite-dimensional superalgebras included) with respect to which the multiplication is continuous. 2) Such norms cannot be chosen to be submultiplicative and equal to one on the unit of the algebra.

SUPERSYMMETRIC RACAH BASIS, FAMILY OF INFINITE-DIMENSIONAL SUPERALGEBRAS, SU(∞+1|∞) AND RELATED 2D MODELS

The irreducible Racah basis for SU(N+1|N) is introduced. An analytic continuation with respect to N leads to infinite-dimensional superalgebras su(v+1|v). The large-v limit su(∞+1|∞) is calculated. A higher spin Sugawara construction leading to generalizations of the Virasoro algebra with infinite tower of higher spin currents is proposed, and the related WZNW and Toda models as well as the possible applications in string theory are discussed.

QUANTIZATION AND COCYCLES ON THE SUPERTORUS AND LARGE-N LIMITS FOR THE CLASSICAL LIE SUPERALGEBRAS

The Poisson superbracket Lie superalgebra on the supertorus T2d|N is considered and its quantization is carried out. It is shown that there exists a non-trivial supercentral extension by means of 2d arbitrary c-numbers (when N is even), or 2d Grassmann numbers (when N is odd). It is shown that the infinite-dimensional superalgebras on the supertorus T2d|N can be considered as certain generalizations and large-M limits of the classical superalgebras A(M| M) and Q(M) (when N is even and odd respectively).

A superconformal theory of massless higher spin fields in D = 2 + 1

We construct a superconformal theory of higher spin fields in a space-time of dimension D = 2 + 1. The construction relies on the infinite-dimensional superalgebra shsc( N | 3) with the superconformal algebra osp( N | 4) as a maximal finite-dimensional subalgebra. The invariant Chern-Simons action for the higher spin superconformal theory is an extension of the usual conformal supergravity action for particles with maximal spin two. The quantization was carried out and the generating functional of the Green functions was obtained.

General D=1 local supercoordinate transformations and their supercurrent algebras

The infinite-dimensional superalgebras W N , describing arbitrary local supercoordinate transformations in D=1 N-extended superspace, are considered as local W N supercurrent algebras. For N=1 and N=2 these superalgebras admit the central extension, necessary for their quantum realizations. The W 1 supercurrents can be identified with N=2 conformal supercurrents, and the W 2 supercurrent algebra is the extension of the SU(2), N=4 conformal supercurrent algebras.

Chern-Simons theories of symplectic super-diffeomorphisms

We discuss the symplectic diffeomorphisms of a class of supermanifolds and the structure of the underlying infinite dimensional superalgebras. We construct a Chern-Simons (CS) gauge theory in (2+1) dimensions for these algebras. There exists a finite dimensional supersymmetric truncation which is the (2 n − 1)-dimensional hamiltonian superalgebra H ̃ (n) . With a central charge added, it is a superalgebra, C(n) , associated with a Clifford algebra. We find an embedding of d=3, N=2 anti-de Sitter superalgebra OSp(2|2) ⊗ OSp (2|2) in C (4), and construct a CS action for its infinite dimensional extension. We also discuss the construction fdof a CS action for the infinite dimensional extension of the d=3, N=2 superconformal algebra OSp(2, 4).

New N = 6 infinite-dimensional superalgebra with central extension

We consider doubly infinite superalgebras describing the D = 2 field theories with locally gauged super Virasoro algebras. Our superalgebras belong to the class of generalized loop algebras S → g1|N, where g1|N is the infinite-dimensional alge bra of superdiffeomorphisms of the supercircle S1|N. It appears that among such doubly infinite superalgebras only the one with g1|N=KNS(1|N), where N=6 (N=6 Neveu-Schwarz conformal superalgebra), admits a central extension due to the existence of a Killing form. Possible applications are outlined.

A SUPERCONFORMAL THEORY OF MASSLESS HIGHER SPIN FIELDS IN D=2+1

We construct a superconformal theory of higher spin fields in a space-time of dimension D=2+1. The construction relies on the infinite-dimensional superalgebra shsc (N/3) with the superconformal algebra OSp (N/4) as a maximal finite-dimensional subalgebra. The invariant Chern-Simons action for the higher spin superconformal theory is an extension of the usual conformal supergravity action for particles with maximal spin two.

A consistent interacting massless higher-spin field theory in D=2+1

Using the approach of Fradkin and Vasiliev (1987), an action is constructed in 2+1 spacetime dimensions describing interacting massless fields of all integer and half-integer spins s>or=3/2. The action is associated with an infinite-dimensional superalgebra, denoted shs(1, 2)(+)shs(1,2). Truncation to the spin 3/2-spin 2 sector gives the (1,1) type anti-de Sitter (AdS) supergravity theory corresponding to osp(1,2; R)(+)osp(1,2; R). Various properties of the D=3 higher-spin theory, and its relevance to the higher-spin problem in four dimensions, are discussed.

Equations of motion of interacting massless fields of all spins as a free differential algebra

It is argued that the equations of motion of interacting massless fields of all spins s=0,1,…,∞ can naturally be formulated in terms of a free differential algebra (FDA) constructed from one-forms and zero-forms that belong both to the adjoint representation of the infinite-dimensional superalgebra of higher spins and auxiliary fields proposed previously. This FDA is found explicitly in the first non-trivial order in the zero-forms. Various properties of the proposed FDA are discussed including the ways for incorporating internal (Yang-Mills) gauge symmetries via associative algebras.