Contraction-based classification of supersymmetric extensions of kinematical lie algebras
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ELEMENTARY PARTICLES AND FIELDS Theory
Contraction-Based Classification of Supersymmetric Extensions of Kinematical Lie Algebras* R. Campoamor-Stursberg1)** and M. Rausch de Traubenberg2) Received April 17, 2009
Abstract—We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra ´ leads to a hierarchy similar in structure to the classical Bacry–Levy-Leblond classification. DOI: 10.1134/S1063778810020109
1. INTRODUCTION The contraction approach has been systematically applied in physics, among other problems, to classify the possible classical kinematical groups, basing on space isotropy and assuming that time-reversal and parity are automorphisms of the kinematical group, as well as noncompactness of one-parameter subgroups generated by boosts [1]. Within this frame, all kinematical models arise as contractions of the de Sitter Lie algebras. It is therefore natural to ask whether for the supersymmetric extensions of the latter algebras, constructed in supersymmetric models, a similar procedure and classification holds, at least for those extensions proven to be of physical interest. The main objective of this work is to extend the classical kinematical classification of Bacry and ´ Levy-Leblond (BBL classification) to the supersym¨ u–Wigner ¨ metric case, using generalized Inon contractions. By means of this procedure, we show that supersymmetric extensions of kinematical algebras considered in the literature [2, 3] fit into a contraction scheme. Contractions of supersymmetric extensions have usually been considered separately, for the most relevant cases [4–6]. A BBL classification however allows us to treat non-standard models like Carroll and Newton algebras in a unified manner. This provides an alternative perspective to the numerous works developed in connection to nonrelativistic limits of supersymmetric theories [5, 7]. We briefly recall the notion of contraction. Given a k over a fixed baLie algebra g with structure tensor Cij sis {Xi } , i = 1, . . . , n, a linear redefinition of the gen∗
The text was submitted by the authors in English. IMI, Universidad Complutense de Madrid, Spain. 2) Universite´ Louis Pasteur, Strasbourg, France. ** E-mail: [email protected] 1)
erators via a matrix A ∈ GL(n, R) gives the transformed structure tensor n m = Ai k Aj (A−1 )m n Ck . Cij
(1)
Considering a family Φ ∈ GL(n, R) of nonsingular linear maps of g, where ∈ (0, 1], for any X, Y ∈ g we define [X, Y ]Φ := Φ−1 [Φ (X), Φ (Y )] ,
(2)
which obviously reproduces the brackets of the Lie algebra over the transformed basis. Actually, this is nothing but equation (1) for a special kind of transformations. Now suppose that the limit [X, Y ]∞ := lim Φ−1 [Φ (X), Φ (Y )] →0
(3)
exists for any X, Y ∈ g. Then equation (3) defines a Lie algebra g which is a contraction of g, since it corresponds to a limiting point of the orbit. We say that the contraction is nontrivial if g and g
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