Generalized Wigner-Inonu Contractions and Maxwell (Super)Algebras
Jerzy Lukierski
备注：
10p; to appear in Proc. of Intern. Conference "Gauge Fields. Yesterday, Today, Tomorrow" dedicated to the 70-th birthday of Andrei A. Slavnov. Steklov Mathematical Institute, Moscow, Russia, January 2010; to be published in Proceedings of Steklov Institute of Russian Acad. of Science
We consider a class of generalized Inonu-Wigner contraction for semidirect product of two particularly related semisimple Lie (super)algebras. The special class of such contractions provides D=4 Maxwell algebra and recently introduced simple D=4 Maxwell superalgebra. Further we present two types of D=4 N-extended Maxwell superalgebras, the nonstandard one for any N with 1/2 N(N-1) central charges and the standard one, for even N=2k, with k(2k-1) internal symmetry generators.
We examine a Wess-Zumino term, written in bilinear of superinvariant currents, for a superstring in anti-de Sitter (AdS) space. The standard Inonu-Wigner contraction does not give the correct flat limit but gives zero. This originates from the fact that the fermionic metric of the super-Poincare group is degenerate. We propose a generalization of the Inonu-Wigner contraction which reduces the super-AdS group to the "nondegenerate" super-Poincare group, therefore it gives a correct flat limit of this Wess-Zumino term. We also discuss the M-algebra obtained by this generalized Inonu-Wigner contr…
We discuss Inönü-Wigner contractions of affine Kac-Moody algebras. We show that the Sugawara construction for the contracted affine algebra exists only for a fixed value of the level $k$, which is determined in terms of the dimension of the uncontracted part of the starting Lie algebra, and the quadratic Casimir in the adjoint representation. Further, we discuss contractions of $G/H$ coset spaces, and obtain an affine {\it translation} algebra, which yields a Virasoro algebra (via a GKO construction) with a central charge given by $dim(G/H)$.
We present a new method to find solutions of the Virasoro master equations for any affine Lie algebra $\widehat{g}$. The basic idea is to consider first the simplified case of an Inönü-Wigner contraction $\widehat{g}_c$ of $\widehat{g}$ and to extend the Virasoro constructions of $\widehat{g}_c$ to $\widehat{g}$ by a perturbative expansion in the contraction parameter. The method is then applied to the orthogonal algebras, leading to fixed-level multi-parameter Virasoro constructions, which are the generalisations of the one-parameter Virasoro construction of $\widehat{su}(2)$ at level four.
We present a generalization of the standard Inönü-Wigner contraction by rescaling not only the generators of a Lie superalgebra but also the arbitrary constants appearing in the components of the invariant tensor. The procedure presented here allows to obtain explicitly the Chern-Simons supergravity action of a contracted superalgebra. In particular we show that the Poincaré limit can be performed to a $D=2+1$ $\left(p,q\right) $ $% AdS$ Chern-Simons supergravity in presence of the exotic form. We also construct a new three-dimensional $\left(2,0\right) $ Maxwell Chern-Simons supergravity theo…
Einstein's $E = mc^{2}$ unifies the momentum-energy relations for massive and massless particles. According to Wigner, the internal space-time symmetries of massive and massless particles are isomorphic to $O(3)$ and $E(2)$ respectively. According to Inonu and Wigner, $O(3)$ can be contracted to $E(2)$ in the large-radius limit. It is noted that the $O(3)$-like little group for massive particles can be contracted to the $E(2)$-like little group for massless particles in the limit of large momentum and/or small mass. It is thus shown that transverse rotational degrees of freedom for massive par…
We construct consistent Kaluza-Klein truncations of type IIA supergravity on (i) $Σ_2\times S^3$ and (ii) $Σ_3\times S^3$, where $Σ_2 = S^2/Γ$, $\mathbb{R}^2/Γ$, or $\mathbb{H}^2/Γ$, and $Σ_3 = S^3/Γ$, $\mathbb{R}^3/Γ$, or $\mathbb{H}^3/Γ$, with $Γ$ a discrete group of symmetries, corresponding to NS5-branes wrapped on $Σ_2$ and $Σ_3$. The resulting theories are a $D=5$, $\mathcal{N}=4$ gauged supergravity coupled to three vector multiplets with scalar manifold $SO(1,1)\times SO(5,3)/(SO(5)\times SO(3))$ and gauge group $SO(2)\times\left(SO(2)\ltimes_{Σ_2}\mathbb{R}^4\right)$ which depends on …
The way to obtain massive non-relativistic states from the Poincare algebra is twofold. First, following Inonu and Wigner the Poincare algebra has to be contracted to the Galilean one. Second, the Galilean algebra is to be extended to include the central mass operator. We show that the central extension might be properly encoded in the non-relativistic contraction. In fact, any Inonu-Wigner contraction of one algebra to another, corresponds to an infinite tower of abelian extensions of the latter. The proposed method is straightforward and holds for both central and non-central extensions. Apa…
We analyze under which conditions the missing label problem associated to a reduction chain $\frak{s}^{\prime}\subset \frak{s}$ of (simple) Lie algebras can be completely solved by means of an Inönü-Wigner contraction $\frak{g}$ naturally related to the embedding. This provides a new interpretation of the missing label operators in terms of the Casimir operators of the contracted algebra, and shows that the available labeling operators are not completely equivalent. Further, the procedure is used to obtain upper bounds for the number of invariants of affine Lie algebras arising as contractions…
The development of the notion of group contraction first introduced by E. In{ö}n{ü} and E.P. Wigner in 1953 is briefly reviewed. The fundamental role of the idea of degenerate transformations is stressed. The significance of contractions of algebraic structures for exactly solvable problems of mathematical physics is noticed.
Maximally supersymmetric spacetime algebras in eleven-dimensions, which are the isometry superalgebras of Minkowski space, AdS_7 x S^4, AdS_4 x S^7 and pp-wave background, are related by Inonu-Wigner contractions. The super-AdS_{4(7)} x S^{7(4)} algebras allow to introduce two contraction parameters, the one for the flat limit to the super-Poincare algebra and the other for a Penrose limit to the super-pp-wave algebra. Under these contractions supersymmetries are maintained because the Jacobi identity of three supercharges holds for any values of contraction parameters.
The isometry algebras of the maximally supersymmetric solutions of IIB supergravity are derived by the Inonu-Wigner contractions of the super-AdS_5xS^5 algebra. The super-AdS_5xS^5 algebra allows introducing two contraction parameters; the one for the Penrose limit to the maximally supersymmetric pp-wave algebra and the AdS_5xS^5 radius for the flat limit. The fact that the Jacobi identity of three supercharges holds irrespectively of these parameters reflects the fact that the number of supersymmetry is not affected under both contractions.
The method of graded contractions, based on the preservation of the automorphisms of finite order, is applied to the affine Kac-Moody algebras and their representations, to yield a new class of infinite dimensional Lie algebras and representations. After the introduction of the horizontal and vertical gradings, and the algorithm to find the horizontal toroidal gradings, I discuss some general properties of the graded contractions, and compare them with the Inönü-Wigner contractions. The example of $\hat A_2$ is discussed in detail.
In this thesis Chern-Simons theories based on Lie algebras with invariant metric are constructed. It is discussed how contractions lead systematically to (higher spin) kinematical algebras of, e.g., Poincaré, Galilei and Carroll type and how they can be extended to permit an invariant metric. A new generalization of the Inönü-Wigner contractions explains why the invariant metric stays nondegenerate in the ultra-relativistic limes to the Carroll algebras. Since asymptotic symmetries give a first glance of possible dual theories consistent boundary conditions for Lifshitz and null-warped higher …
According to the literature, the S-expansion procedure involving a finite semigroup is valid no matter what the structure of the original Lie (super)algebra is; However, when something about the structure of the starting (super)algebra is known and when certain particular conditions are met, the S-expansion method (with its features of resonance and reduction) is able not only to lead to several kinds of expanded (super)algebras, but also to reproduce the effects of the standard as well as the generalized Inönü-Wigner contraction. In the present paper, we propose a new prescription for S-expan…
We consider how the continuous spin representation (CSR) of the Poincare group in four dimensions can be generated by dimensional reduction. The analysis uses the front-form little group in five dimensions, which must yield the Euclidean group E(2), the little group of the CSR. We consider two cases, one is the single spin massless representation of the Poincare group in five dimensions, the other is the infinite component Majorana equation, which describes an infinite tower of massive states in five dimensions. In the first case, the double singular limit j,R go to infinity, with j/R fixed, w…
We consider the curvatures 2 form asociated with AdSL4 valued one-form gauge connetion, and then we construct a four-dimensional action that generalize the Einstein-Hilbert gravity. It is shown that the Maxwell extension of Einstein gravity can be obtained from AdSL4-gravity making use of the Inonu-Wigner contraction method. In the same way, by gauging the AdSL5 spacetime algebra, the Einstein-Hilbert gravity is extended including the vector fields kab and ha which are associated with non-Abelian tensors and non Abelian vectors charges in the AdSL5 algebra. The B5 extension of Einstein gravity…
In this paper, we discuss the Inönü-Winger contraction of the conformal algebra. We start with the light-cone form of the Poincaré algebra and extend it to write down the conformal algebra in $d$ dimensions. To contract the conformal algebra, we choose five dimensions for simplicity and compactify the third transverse direction in to a circle of radius $R$ following Kaluza-Klein dimensional reduction method. We identify the inverse radius, $1/R$, as the contraction parameter. After the contraction, the resulting representation is found to be the continuous spin representation in four dimension…
Galilean conformal algebra (GCA) is an Inonu-Wigner (IW) contraction of a conformal algebra, while Newton-Hooke string algebra is an IW contraction of an AdS algebra which is the isometry of an AdS space. It is shown that the GCA is a boundary realization of the Newton-Hooke string algebra in the bulk AdS. The string lies along the direction transverse to the boundary, and the worldsheet is AdS_2. The one-dimensional conformal symmetry so(2,1) and rotational symmetry so(d) contained in the GCA are realized as the symmetry on the AdS_2 string worldsheet and rotational symmetry in the space tran…
We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order three. All these algebras are related through generalised Inonü-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order three.
We introduce in this paper the contractions $\mathfrak{G}_c$ of $n$-Lie (or Filippov) algebras $\mathfrak{G}$ and show that they have a semidirect structure as their $n=2$ Lie algebra counterparts. As an example, we compute the non-trivial contractions of the simple $A_{n+1}$ Filippov algebras. By using the İnönü-Wigner and the generalized Weimar-Woods contractions of ordinary Lie algebras, we compare (in the $\mathfrak{G}=A_{n+1}$ simple case) the Lie algebras Lie$\,\mathfrak{G}_c$ (the Lie algebra of inner endomorphisms of $\mathfrak{G}_c$) with certain contractions $(\mathrm{Lie}\,\mathfrak…
After reviewing the three well-known methods to obtain Lie algebras and superalgebras from given ones, namely, contractions, deformations and extensions, we describe a fourth method recently introduced, the expansion of Lie (super)algebras. Expanded (super)algebras have, in general, larger dimensions than the original algebra, but also include the Inonu-Wigner and generalized IW contractions as a particular case. As an example of a physical application of expansions, we discuss the relation between the possible underlying gauge symmetry of eleven-dimensional supergravity and the superalgebra o…
The non-relativistic versions of the generalized Poincaré algebras and generalized $AdS$-Lorentz algebras are obtained. This non-relativistic algebras are called, generalized Galilean algebras type I and type II and denoted by $\mathcal{G}\mathfrak{B}_{n}$ and $\mathcal{G}\mathfrak{L}_{_{n}}$ respectively. Using a generalized Inönü--Wigner contraction procedure we find that the generalized Galilean algebras type I can be obtained from the generalized Galilean algebras type II. The $S$-expansion procedure allows us to find the $\mathcal{G}\mathfrak{B}_{_{5}}$ algebra from the Newton--Hooke alge…
F-Lie algebras are natural generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). We give finite dimensional examples of F-Lie algebras obtained by an inductive process from Lie algebras and Lie superalgebras. Matrix realizations of the $F-$Lie algebras constructed in this way from osp(2|m) are given. We obtain a non-trivial extension of the Poincaré algebra by an Inönü-Wigner contraction of a certain $F-$Lie algebras with $F>2$.
In this paper we briefly review the main results obtained in arXiv:0812.1982, where some algebraic properties of the 'vector supersymmetry' (VSUSY) algebra have been studied. VSUSY is a graded extension of the Poincare' algebra in 4 dimensions with two central charges. We derive all independent Casimir operators of VSUSY and we find two distinct spin-related operators in the case of nonvanishing central charges. One is the analogue of superspin for VSUSY and the other is a new spin, called C-spin, whose value is fixed to 1/2. We also show that the VSUSY algebra and its Casimir operators can be…
Any $(d+1)$-dimensional CFT with a $U(1)$ flavor symmetry, a BPS bound and an exactly marginal coupling admits a decoupling limit in which one zooms in on the spectrum close to the bound. This limit is an Inönü-Wigner contraction of $so(2,d+1)\oplus u(1)$ that leads to a relativistic algebra with a scaling generator but no conformal generators. In 2D CFTs, Lorentz boosts are abelian and by adding a second $u(1)$ we find a contraction of two copies of $sl(2,\mathbb{R})\oplus u(1)$ to two copies of $P_2^c$, the 2-dimensional centrally extended Poincaré algebra. We show that the bulk is described…
$F-$Lie algebras are natural generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). When $F>2$ not many finite-dimensional examples are known. In this paper we construct finite-dimensional $F-$Lie algebras $F>2$ by an inductive process starting from Lie algebras and Lie superalgebras. Matrix realisations of $F-$Lie algebras constructed in this way from $\mathfrak{su}(n), \mathfrak{sp}(2n)$ $\mathfrak{so}(n)$ and $\mathfrak{sl}(n|m)$, $\mathfrak{osp}(2|m)$ are given. We obtain non-trivial extensions of the Poincaré algebra by Inönü-Wigner contraction of certain $F-$Lie algebras with…
Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, are intrinsically associated to null manifolds and in two dimensions can be obtained as an In{ö}n{ü}-Wigner contraction of the two-dimensional ($2d$) relativistic conformal algebra. Instead of performing contractions, we demonstrate in this paper how this transmutation of symmetries can be achieved by infinite boosts or degenerate linear transformations on coordinates. Taking explicit cues from the worldsheet theory of null strings, we show boosting the system is equivalent to adding a current-current deformati…
Contractions (and graded contractions) of Lie algebra, Lie bialgebra and Hopf algebra are discussed. It is noticed the fundamental role of E.In{ö}n{ü} and E.P.Wigner idea of degenerate transformations. A constructive algorithm for description of contractions of quantum Cayley-Klein algebras $ so_{z}(n+1; {\bf j}) $ with different choice of the set of primitive operators is suggested. For nonsemisimple quantum algebras it gives nonisomorphic Hopf algebras. From physical point of view this algorithm gives the different physical interpretations of primitive operators for mathematically the same n…
Recently was shown that standard odd and even-dimensional General Relativity can be obtained from a $(2n+1)$-dimensional Chern-Simons Lagrangian invariant under the $B_{2n+1}$ algebra and from a $(2n)$-dimensional Born-Infeld Lagrangian invariant under a subalgebra $\cal{L}^{B_{2n+1}}$ respectively. Very Recently, it was shown that the generalized Inönü-Wigner contraction of the generalized AdS-Maxwell algebras provides Maxwell algebras types $\cal{M}_{m}$ which correspond to the so called $B_{m}$ Lie algebras. In this article we report on a simple model that suggests a mechanism by which stan…
We discuss the issue of how to include magnetic charges in the AdS(4) superalgebra osp(4|2). It is shown that the usual way of introducing a pseudoscalar central charge on the right hand side of the basic anticommutator does not work, because this breaks SO(2,3) covariance. We propose a way out by promoting the magnetic charge to a vector charge, which amounts to enlarge osp(4|2) to the superconformal algebra su(2,2|1). The conditions for 1/4, 1/2 and 3/4 BPS states are then analyzed. These states form the boundary of the convex cone associated with the Jordan algebra of 4x4 complex hermitian …
The semisimple part of d-dimensional Galilean conformal algebra g^(d) is given by h^(d)=O(2,1)+O(d), which after adding via semidirect sum the 3d-dimensional Abelian algebra t^(d) of translations, Galilean boosts and constant accelerations completes the construction. We obtain Galilean superconformal algebra G^(d) by firstly defining the semisimple superalgebra H^(d) which supersymmetrizes h^(d), and further by considering the expansion of H^(d) by tensorial and spinorial graded Abelian charges in order to supersymmetrize the Abelian generators of t^(d). For d=3 the supersymmetrization of h^(3…
It is noted that two-by-two S-matrices in multilayer optics can be represented by the Sp(2) group whose algebraic property is the same as the group of Lorentz transformations applicable to two space-like and one time-like dimensions. It is noted also that Wigner's little groups have a sliderule-like property which allows us to perform multiplications by additions. It is shown that these two mathematical properties lead to a cyclic representation of the S-matrix for multilayer optics, as in the case of ABCD matrices for laser cavities. It is therefore possible to write the N-layer S-matrix as a…
We study how to generate new Lie algebras $\mathcal{G}(N_0,..., N_p,...,N_n)$ from a given one $\mathcal{G}$. The (order by order) method consists in expanding its Maurer-Cartan one-forms in powers of a real parameter $λ$ which rescales the coordinates of the Lie (super)group $G$, $g^{i_p} \to λ^p g^{i_p}$, in a way subordinated to the splitting of $\mathcal{G}$ as a sum $V_0 \oplus ... \oplus V_p \oplus ... \oplus V_n$ of vector subspaces. We also show that, under certain conditions, one of the obtained algebras may correspond to a generalized İnönü-Wigner contraction in the sense of Weimar-W…
