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Chevalley Supergroups of Type D(2, 1; a)

Published online by Cambridge University Press:  21 August 2013

F. Gavarini
F. Gavarini*
Affiliation:
Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Roma, Italy, (gavarini@mat.uniroma2.it)
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Abstract

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We present a construction ‘à la Chevalley’ of connected affine supergroups associated with Lie superalgebras of type D(2, 1; a), for any possible value of the parameter a. This extends the results by Fioresi and Gavarini, in which all other simple Lie superalgebras of classical type were considered. The case of simple Lie superalgebras of Cartan type is dealt with in a previous paper by the author, so this work completes the programme of constructing connected affine supergroups associated with any simple Lie superalgebra.

Keywords

simple Lie superalgebrasaffine supergroupsrepresentation of Lie superalgebrasPrimary 14M3014A22Secondary 17B20

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 465 - 491
Copyright
Copyright © Edinburgh Mathematical Society 2014 

References

1.Bahturin, Y. A., Mikhalev, A. A., Petrogradsky, V. M. and Zaicev, M. V., Infinite-dimensional Lie superalgebras, De Gruyter Expositions in Mathematics, Volume 7 (Walter de Gruyter, Berlin, 1992).CrossRefGoogle Scholar
2.Brundan, J. and Kleshchev, A., Modular representations of the supergroup Q(n), I, J. Alg. 206 (2003), 64–98.Google Scholar
3.Brundan, J. and Kujava, J., A new proof of the Mullineux conjecture, J. Algebraic Combin. 18 (2003), 13–39.Google Scholar
4.Carmeli, C., Caston, L. and Fioresi, R., Mathematical foundations of supersymmetry, European Mathematical Society Series of Lectures in Mathematics, Volume 15 (European Mathematical Society, Zurich, 2011).CrossRefGoogle Scholar
5.Fioresi, R. and Gavarini, F., On the construction of Chevalley supergroups, in Super-symmetry in mathematics and physics, pp. 101–123, Lecture Notes in Mathematics, Volume 2027 (Springer, 2011).Google Scholar
6.Fioresi, R. and Gavarini, F., Chevalley supergroups, Memoirs of the American Mathematical Society, Volume 215 (American Mathematical Society, Providence, RI, 2012).Google Scholar
7.Frappat, L., Sorba, P. and Sciarrino, A., Dictionary on Lie algebras and superalgebras (Academic, 2000).Google Scholar
8.Gavarini, F., Algebraic supergroups of Cartan type, Forum Math., DOI:10.1515/forum-2011-0144.Google Scholar
9.Humphreys, J. E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Volume 9 (Springer, 1972).Google Scholar
10.Iohara, K. and Koga, Y., Central extensions of Lie superalgebras, Comment. Math. Helv. 76 (2001), 110–154.CrossRefGoogle Scholar
11.Kac, V. G., Lie superalgebras, Adv. Math. 26 (1977), 8–26.Google Scholar
12.Masuoka, A., The fundamental correspondences in super affine groups and super formal groups, J. Pure Appl. Alg. 202 (2005), 284–312.Google Scholar
13.Scheunert, M., The theory of Lie superalgebras, Lecture Notes in Mathematics, Volume 716 (Springer, 1979).Google Scholar
14.Shu, B. and Wang, W., Modular representations of the ortho-symplectic supergroups, Proc. Lond. Math. Soc. 96 (2008), 251–271.CrossRefGoogle Scholar
15.Varadarajan, V. S., Supersymmetry for mathematicians: an introduction, Courant Lecture Notes, Volume 1 (American Mathematical Society, Providence, RI, 2004).Google Scholar