Abstract
The concept of Grassmannification of a Lie group, which is completely analogous to the concept of complexification of a Lie group, is introduced. Grassmannified Lie groups can also be viewed as ordinary real Lie groups. It is shown that every graded Lie algebra (= superalgebra) determines a subgroup (Kac-Berezin group, supergroup) in the Grassmannified full matrix group. On the other hand, it seems possible that not all supergroups can be found by a complete classification of all graded Lie algebras.
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References
Corwin, L., Ne'eman, Y., and Sternberg, S. (1975).Rev. Mod. Phys.,47, 573.
Nahm, W., and Scheunert, M. (1976).J. Math. Phys.,17, 868.
Pais, A., and Rittenberg, V. (1975).J. Math. Phys.,16, 2062.
Nahm, W. (1978).Nucl. Phys.,B135, 149.
Berezin, F. A., and Marinov, M. S. (1977).Ann. Phys. (N.Y.),104, 336.
Berezin, F. A. (1966).The Method of Second Quantization, Academic Press, New York
Berezin, F. A., and Kac, G. (1970).Mat. Sb. (SSSR),82, 124 [English transl. (1970).Math. USSR Sbornik,11, 311].
Fayet, P., and Ferrara, S. (1977).Phys. Rep. C,32, 249.
Van Nieuwenhuizen, P. (1980). InCosmology and Gravitation, ed. P. Bergman and V. De Sabbata, Plenum Press, New York, p. 227.
Nieuwenhuizen, P. van, (1981). InSuperspace and Supergravity, Proceedings of the Nuffiled Workshop, Cambridge U.P., Cambridge, p. 9.
Rocek, M., (1981).Superspace and Supergravity, Cambridge U.P., p. 71.
Freedman, D. Z., and Nieuwenhuizen, P. van, (1979). InSupergravity, North-Holland, Amsterdam.
Freedman, D. Z., Nieuwenhuizen, P. van, and Ferrara, S. (1976).Phys. Rev. D,13, 3214.
Deser, S., and Zumino, B. (1976).Phys. Lett.,62B, 335.
Ferrara, S., and Nieuwenhuizen, P. van, (1976).Phys. Rev. Lett.,37, 1669.
Ferrara, S., Scherk, J., and Zumino, B. (1977).Phys. Lett.,66, 35; (1977).Nucl. Phys.,B121, 393.
Freedman, D. Z. (1977).Phys. Rev. Lett.,38, 105.
Das, A. (1977).Phys. Rev. D,15, 2805.
Cremmer, I. Scherk, J., and Ferrara, S. (1977).Phys. Lett.,68B, 234.
Cremmer, E., and Scherk, J. (1977).Nucl. Phys.,B127, 259.
De Wit, B., and Freedman, D. Z. (1977).Nucl Phys.,B130, 105.
Wess, J., and Zumino, B. (1974).Nucl. Phys.,B70, 39.
Berezin, F. A., and Kac, G. I. (1970).Mat. Sb. (SSSR),82, 311 [English transl. (1970).Math. USSR Sbornik,11, 124].
Abraham, R., and Marsden, J. E. (1978).Foundations of Mechanics, Benjamin, New York.
Chevalley, C. (1946).Theory of Lie Groups, Princeton U.P., Princeton, New Jersey.
Pontryagin, L. S. (1966).Topological Groups, Chaps. 7–10, Gordon & Breach, New York.
van der Waerden, B. L. (1974).Group Theory and Quantum Mechanics, Springer-Verlag, Berlin.
Boerner, H. (1970).Representations of Groups, Holland Publ. Company, Amsterdam.
DeWitt, B. S. (1964). “Dynamical Theory of Groups and Fields,” in DeWitt and DeWitt,Relativity, Groups and Topology, Gordon & Breach, New York.
Freedman, D. Z., and van Nieuwanhuizen, P. (1976).Phys. Rev. D,14, 912.
Goddard, P. (1975).Nucl. Phys.,B88, 429.
Hestenes, D. (1966).Space-Time Algebra, Gordon and Breach, New York.
Gilmore, R. (1974).Lie Groups, Lie Algebras, and Some of Their Applications, John Wiley, New York.
Scheunert, M. (1979).The Theory of Lie Superalgebras, Lecture Notes in Mathematics 716, Springer-Verlag, Berlin.
Nieuwenhuizen, P. van, (1981). “Supergravity,”Phys. Rep. C,68, 189.
Cartan, Elie (1966).The Theory of Spinors, Hermann, Paris.
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Ebner, D.W. The supergroups of supersymmetry and supergravity as ordinary real Lie groups. Gen Relat Gravit 14, 1001–1016 (1982). https://doi.org/10.1007/BF00756282
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DOI: https://doi.org/10.1007/BF00756282