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On the Relation of Real and Complex Lie Supergroups

Published online by Cambridge University Press:  20 November 2018

Matthias Kalus
Matthias Kalus*
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraβe 150, D- 44801 Bochum, Germany. e-mail: matthias.kalus@rub.de
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Abstract

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A complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie supergroup are deduced. The classification of real Lie supergroups with such almost complex structures yields a new approach to the known classification of complex Lie supergroups by complexHarish-Chandra superpairs. A universal complexification of a real Lie supergroup is constructed.

Keywords

32C1158A50Lie supergroupalmost complex structureHarish–Chandra pairuniversalcomplexification

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 281 - 284
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Berezin, F. A., Introduction to superanalysis. Mathematical Physics and Applied Mathematics, 9, Reidel Publishing Co., Dordrecht, 1987 Google Scholar
[2] Deligne, P. and Morgan, J. W., Notes on supersymmetry (following Joseph Bernstein). In: Quantum fields and strings: a course for mathematicians, vol.1, American Mathematical Society, Providence, RI, 1999, pp. 41–97.Google Scholar
[3] Hochschild, G., The structure of Lie groups. Holden-Day, Inc., San Francisco-London-Amsterdam, 1965.Google Scholar
[4] Kalus, M., Complex analytic aspects of Lie Supergroups. Dissertation, RuhrUniversitât Bochum, Bochum, Germany, 2011.Google Scholar
[5] Kobayashi, S. and Nomizu, K., Foundations of differential geometry. II. Wiley-Interscience, 1969.Google Scholar
[6] Kostant, B., Graded manifolds, graded Lie theory, andprequantization. In: Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Lecture Notes in Math,. 570, Springer, Berlin, 1977, pp. 177–306.Google Scholar
[7] McHugh, A., A Newlander-Nirenberg theorem for supermanifolds. I. Math. Phys. 30 (1989), no. 5, 1039–1042. http://dx.doi.Org/10.1063/1.528373 Google Scholar
[8] Vaïntrob, A. Yu., Almost complex structures on supermanifolds. In: Problems in group theory and homological algebra (Russian), Yaroslav. Gos. Univ., Yaroslavl', 1985, pp. 139–142,166.Google Scholar
[9] Vishnyakova, E. G., On complex Lie supergroups and split homogeneous supermanifolds. Transform. Groups 16 (2011), no. 1, 265–285. http://dx.doi.Org/10.1007/s00031-010-9114-5 Google Scholar