Vector supersymmetry: Casimir operators and contraction from Ø Sp (3,2 |2) (original) (raw)
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Vector supersymmetry from OSp(3,2|2): Casimir operators
Fortschritte der Physik, 2009
In this paper we briefly review the main results obtained in , where some algebraic properties of the 'vector supersymmetry' (VSUSY) algebra have been studied. VSUSY is a graded extension of the Poincaré 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 derived by an Inönü-Wigner contraction from OSp(3, 2 |2). This paper is based on the talk given in Varna, Bulgaria, during the 4-th EU RTN Workshop 2008.
Quasideterminants and Casimir elements for the general Lie superalgebra
arXiv (Cornell University), 2003
We apply the techniques of quasideterminants to construct new families of Casimir elements for the general linear Lie superalgebra gl(m|n) whose images under the Harish-Chandra isomorphism are respectively the elementary, complete and power sums supersymmetric functions.
Quasideterminants and Casimir elements for the general linear Lie superalgebra
International Mathematics Research Notices, 2004
We apply the techniques of quasideterminants to construct new families of Casimir elements for the general linear Lie superalgebra gl(m|n) whose images under the Harish-Chandra isomorphism are respectively the elementary, complete and power sums supersymmetric functions.
2005
In this letter, we recall briefly the generalized Casimir elements of a finite dimensional Lie algebra. We specify those of orders two and three : when the Lie algebra is simple (even semisimple), we begin by normalizing the former (the quadratic), and then we study some actions of the latter (the cubic). In particular, we introduce a graphical formalism, translating rigorously the tensorial calculus. This allows us to prove the main theorem in a graphic theoretic manner. MIRAMARE TRIESTE August 1996 1 E-mail: elhouari@ictp.trieste.it and elhouari@drakkar.ens.fr 1History and Introduction The Casimir elements of a Lie algebra are those in the center of its enveloping algebra. They play a crucial role in representation theory. They are specially useful for both theoretical physicists and mathematicians who work in the frontier of physics, J-P Elliot [?], M. Gell-Mann [?]. In L-C Biedenharn's and Racah's works, [?] and [?] respectively, we notice the use of the Casimir elements...
Supersymmetric Quantum Mechanics and Super-Lichnerowicz Algebras
Communications in Mathematical Physics, 2007
We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p + 2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are tensors or spinor-tensors on the curved background while conserved charges correspond to the various differential geometry operators acting on these. The Hamiltonian generalizes Lichnerowicz's wave/Laplace operator. It is central, and the models are supersymmetric whenever the background is a symmetric space, although there is an osp(2p|Q) superalgebra for any curved background. The lowest purely bosonic example (2p, Q) = (2, 0) corresponds to a deformed Jacobi group and describes Lichnerowicz's original algebra of constant curvature, differential geometric operators acting on symmetric tensors. The case (2p, Q) = (0, 1) is simply the N = 1 superparticle whose supercharge amounts to the Dirac operator acting on spinors. The (2p, Q) = (0, 2) model is the N = 2 supersymmetric quantum mechanics corresponding to differential forms. (This latter pair of models are supersymmetric on any Riemannian background.) When Q is odd, the models apply to spinor-tensors. The (2p, Q) = (2, 1) model is distinguished by admitting a central Lichnerowicz-Dirac operator when the background is constant curvature. The new supersymmetric models are novel in that the Hamiltonian is not just a square of super charges, but rather a sum of commutators of supercharges and commutators of bosonic charges. These models and superalgebras are a very useful tool for any study involving high rank tensors and spinors on manifolds.
Contractions yielding new supersymmetric extensions of the poincaré algebra
Reports on Mathematical Physics, 1991
Two new PoincarC superalgebras are analysed. They are obtained by the Wigner-In6nl contraction from two real forms of the superalgebra OSp(2; 4; C) -one describing the N = 2 anti-de-Sitter superalgebra with a non-compact internal symmetry SO(l, 1) and the other corresponding to the de-Sitter superalgebra with internal symmetry SO(2). Both are 19-dimensional self-conjugate extensions of the Konopel'chenko superalgebra. They contain 10 PoincarC generators and one generator of internal symmetry in addition to 8 odd generators half of which, however, do not commute with translations.
Casimir invariants for the complete family of quasisimple orthogonal algebras
Journal of Physics A-mathematical and General, 1997
A complete choice of generators of the center of the enveloping algebras of real quasi-simple Lie algebras of orthogonal type, for arbitrary dimension, is obtained in a unified setting. The results simultaneously include the well known polynomial invariants of the pseudo-orthogonal algebras so(p,q)so(p,q)so(p,q), as well as the Casimirs for many non-simple algebras such as the inhomogeneous iso(p,q)iso(p,q)iso(p,q), the Newton-Hooke and Galilei type, etc., which are obtained by contraction(s) starting from the simple algebras so(p,q)so(p,q)so(p,q). The dimension of the center of the enveloping algebra of a quasi-simple orthogonal algebra turns out to be the same as for the simple so(p,q)so(p,q)so(p,q) algebras from which they come by contraction. The structure of the higher order invariants is given in a convenient "pyramidal" manner, in terms of certain sets of "Pauli-Lubanski" elements in the enveloping algebras. As an example showing this approach at work, the scheme is applied to recovering the Casimirs for the (3+1) kinematical algebras. Some prospects on the relevance of these results for the study of expansions are also given.
A Clifford algebra realization of Supersymmetry and its Polyvector extension in Clifford Spaces
It is shown explicitly how to construct a novel (to our knowledge) realization of the Poincare superalgebra in 2D. These results can be extended to other dimensions and to (extended) superconformal and (anti) de Sitter superalgebras. There is a fundamental difference between the findings of this work with the other approaches to Supersymmetry (over the past four decades) using Grassmannian calculus and which is based on anti-commuting numbers. We provide an algebraic realization of the anticommutators and commutators of the 2D super-Poincare algebra in terms of the generators of the tensor product Cl1,1(R) ⊗ A of a two-dim Clifford algebra and an internal algebra A whose generators can be represented in terms of powers of a 3 × 3 matrix Q, such that Q 3 = 0. Our realization dif f ers from the standard realization of superalgebras in terms of dif f erential operators in Superspace involving Grassmannian (anticommuting) coordinates θ α and bosonic coordinates x µ . We conclude in the final section with an analysis of how to construct Polyvector-valued extensions of supersymmetry in Clifford Spaces involving spinor-tensorial supercharge generators Q µ 1 µ 2 .....µn α and momentum polyvectors Pµ 1 µ 2 ....µn . Clifford-Superspace is an extension of Clifford-space and whose symmetry transformations are generalized polyvector-valued supersymmetries.
Considerations on Super Poincaré Algebras and Their Extensions to Simple Superalgebras
Reviews in Mathematical Physics, 2002
We consider simple superalgebras which are a supersymmetric extension of the spin algebra in the cases where the number of odd generators does not exceed 64. All of them contain a super Poincaré algebra as a contraction and another as a subalgebra. Because of the contraction property, some of these algebras can be interpreted as de Sitter or anti de Sitter superalgebras. However, the number of odd generators present in the contraction is not always minimal due to the different splitting properties of the spinor representations under a subalgebra. We consider the general case, with arbitrary dimension and signature, and examine in detail particular examples with physical implications in dimensions d = 10 and d = 4.
Field representations of vector supersymmetry
Journal of High Energy Physics, 2010
We study some field representations of vector supersymmetry with superspin Y = 0 and Y = 1/2 and nonvanishing central charges. For Y = 0, we present two multiplets composed of four spinor fields, two even and two odd, and we provide a free action for them. The main differences between these two multiplets are the way the central charge operators act and the compatibility with the Majorana reality condition on the spinors. One of the two is related to a previously studied spinning particle model. For Y = 1/2, we present a multiplet composed of one even scalar, one odd vector and one even selfdual two-form, which is a truncation of a known representation of the tensor supersymmetry algebra in Euclidean spacetime. We discuss its rotation to Minkowski spacetime and provide a set of dynamical equations for it, which are however not derived from a Lagrangian. We develop a superspace formalism for vector supersymmetry with central charges and we derive our multiplets by superspace techniques. Finally, we discuss some representations with vanishing central charges.