Abstract
An analogue of the classical global third Lie theorem is proved to hold for super Lie groups whose ground Banach—Grassmann algebra is (possibly) infinite-dimensional, provided that this algebra has an ordered basis. It is also proved that the superanalytic structure of a connected super Lie group having a prescribed Lie module is unique, although in a weaker sense than in the case of ordinary Lie groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fayet, P. and Ferrara, S., ‘Supersymmetry’, Phys. Reports C32, 249–334 (1977), and references therein.
Rogers, A., ‘A Global Theory of Supermanifolds’, J. Math. Phys. 21, 1352–1365 (1980).
Rogers, A., ‘Super Lie Groups: Global Topology and Local Structure’, J. Math. Phys. 22, 939–945 (1981).
Jadczyk, A. and Pilch, K., ‘Superspaces and Supersymmetries’, Commun. Math. Phys. 78, 373–390 (1981).
Kostant, B., ‘Graded Manifolds, Graded Lie Theory and Prequantization’, in Lecture Notes in Mathematics 570, Springer-Verlag, Berlin, 1977.
Corwin, L., Ne'man, Y., and Sternberg, S., ‘Graded Lie Algebras in Mathematics and Physics (Bose-Fermi Symmetry)’, Rev. Mod. Phys. 47, 573–603 (1975).
Rickart, C.E., General Theory of Banach Algebras, Van Nostrand, Princeton, 1960.
Dieudonné, J., Foundations of Modern Analysis, Academic Press, New York, 1960.
Lang, S., Differential Manifolds, Addison-Wesley, Reading, Mass., 1972.
Cohn, P.M., Lie Groups, Cambridge University Press, Cambridge, 1957.
Cianci, R., ‘Supermanifolds and Automorphisms of Super Lie Groups’, J. Math. Phys. 25, 451–456 (1984); Bruzzo, U. and Cianci, R., Mathematical Theory of Super Fibre Bundles, Classical Quantum Gravity, 1984 (in print).
Kac, V.G., ‘Lie Superalgebras’, Adv. Math. 26, 8–96 (1977).
Van, Est, V.J. and Korthagen, Th.J., ‘Non-enlargible Lie Algebras’, Indag. Math. 26, 15–31 (1964).
Jacobson, N., Lie Algebras, Interscience, New York, 1962.
Singer, I., Bases in Banach Spaces I, Springer-Verlag, Berlin, 1970; Elton Lacey, H., Notes in Banach Spaces, University of Texas Press, Austin and London, 1980.
De la, Harpe, P., ‘Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space’, Lecture Notes in Mathematics 285, Springer-Verlag, Berlin, 1972.
Birkhoff, G., ‘Analytical Groups’, Trans. A.M.S. 43, 61–101 (1938).
Pontrjagin, L., Topological Groups, Princeton University Press, Princeton, NJ, 1958.
Varadarajan, V.S., Lie Groups, Lie Algebras, and Their Representations, Prentice-Hall, Englewood Cliffs, NJ, 1974.
Author information
Authors and Affiliations
Additional information
Work partly supported by the National Group for Mathematical Physics (GNFM) of the Italian National Research Council (CNR), and by the Italian Ministry of Education through the research project ‘Geometry and Physics’.
Rights and permissions
About this article
Cite this article
Bruzzo, U., Cianci, R. An existence result for super Lie groups. Letters in Mathematical Physics 8, 279–288 (1984). https://doi.org/10.1007/BF00400498
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00400498