Abstract
A class of ℤ2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form where the algebra of generalized translations W=W0+W1 is the maximal solvable ideal of W0 is generated by W1 and commutes with W. Choosing W1 to be a spinorial module (a sum of an arbitrary number of spinors and semispinors), we prove that W0 consists of polyvectors, i.e.all the irreducible submodules of W0 are submodules of We provide a classification of such Lie (super)algebras for all dimensions and signatures. The problem reduces to the classification of invariant valued bilinear forms on the spinor module S.
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Alekseevsky, D.V., Cortés, V.: Classification of N-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation of Spin(p, q). Commun. Math. Phys. 183, 477–510 (1997)
de Azcárraga, J.A., Izquierdo, J.M.: Lie groups, Lie algebras, cohomology and some applications in physics. Cambridge: Camb. Univ. Press, 1995
Chryssomalakos, C., de Azcárraga, J.A., Izquierdo, J.M., Pérez Bueno, J.C.: The geometry of branes and extended superspaces. Nucl. Phys. B567, 293–330 (2000)
D’Auria, R., Ferrara, S., Lledo, M.A., Varadarajan, V.S.: Spinor algebras. J. Geom. Phys. 40, 101–129 (2001)
Devchand, C., Nuyts, J.: Supersymmetric Lorentz-covariant hyperspaces and self-duality equations in dimensions greater than (4|4). Nucl. Phys. B503, 627–656 (1997); Lorentz covariant spin two superspaces. Nucl. Phys. B527, 479–498 (1998); Democratic Supersymmetry. J. Math. Phys. 42, 5840–5858 (2001)
Fradkin, E.S., Vasiliev, M.A.: Candidate for the role of higher spin symmetry. Ann. Phys. 177, 63–112 (1987)
Lawson, H.B., Michelson, M.-L.: Spin geometry. Princeton: Princeton University Press, 1989
Nahm, W.: Supersymmetries and their representations. Nucl. Phys. B135 149 (1978)
Okubo, S.: Real representations of finite Clifford algebras: (I) Classification. J. Math. Phys. 32 1657–1668 (1991)
Onishchik, A.L., Vinberg, E.B.: Lie Groups and Algebraic Groups. Berlin-Heidelberg: Springer, 1990
Scheunert, M.: Generalized Lie Algebras. J. Math. Phys. 20 712–720 (1979)
Shnider, S.: The superconformal algebra in higher dimensions. Lett. Math. Phys. 16, 377–383 (1988)
Vasiliev, M.A.: Consistent equations for interacting massless fields of all spins in the first order in curvatures. Ann. Phys. 190, 59–106 (1989)
Van Proeyen, A.: Tools for supersymmetry. Annals of the University of Craiova, Physics AUC 9 (part I), 1–48 (1999)
van Holten, J.W., Van Proeyen, A.: N=1 Supersymmetry Algebras in D = 2, D = 3, D = 4 mod 8. J. Phys. A 15 3763–3783 (1982)
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Communicated by G.W. Gibbons
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Alekseevsky, D., Cortés, V., Devchand, C. et al. Polyvector Super-Poincaré Algebras. Commun. Math. Phys. 253, 385–422 (2005). https://doi.org/10.1007/s00220-004-1155-y
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DOI: https://doi.org/10.1007/s00220-004-1155-y