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SIGMA 14 (2018), 137, 36 pages arXiv:1709.04717
https://doi.org/10.3842/SIGMA.2018.137
Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$
Kenji Iohara a and Fabio Gavarini b
a) Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, F 69622 Villeurbanne Cedex, France
b) Dipartimento di Matematica, Università di Roma ''Tor Vergata'', Via della ricerca scientifica 1, I-00133 Roma, Italy
Received October 31, 2017, in final form December 11, 2018; Published online December 31, 2018
Abstract
The complex Lie superalgebras $\mathfrak{g}$ of type $D(2,1;a)$ - also denoted by $\mathfrak{osp}(4,2;a) $ - are usually considered for ''non-singular'' values of the parameter $a$, for which they are simple. In this paper we introduce five suitable integral forms of $\mathfrak{g}$, that are well-defined at singular values too, giving rise to ''singular specializations'' that are no longer simple: this extends the family of simple objects of type $D(2,1;a)$ in five different ways. The resulting five families coincide for general values of $a$, but are different at ''singular'' ones: here they provide non-simple Lie superalgebras, whose structure we describe explicitly. We also perform the parallel construction for complex Lie supergroups and describe their singular specializations (or ''degenerations'') at singular values of $a$. Although one may work with a single complex parameter $a$, in order to stress the overall $\mathfrak{S}_3$-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter $\boldsymbol{\sigma} = (\sigma_1,\sigma_2,\sigma_3)$ ranging in the complex affine plane $\sigma_1 + \sigma_2 + \sigma_3 = 0$.
Key words:
Lie superalgebras; Lie supergroups; singular degenerations; contractions.
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References
-
Balduzzi L., Carmeli C., Fioresi R., A comparison of the functors of points of supermanifolds, J. Algebra Appl. 12 (2013), 1250152, 41 pages, arXiv:0902.1824.
-
Bouarroudj S., Grozman P., Leites D., Classification of finite dimensional modular Lie superalgebras with indecomposable Cartan matrix, SIGMA 5 (2009), 060, 63 pages, arXiv:0710.5149.
-
Chapovalov D., Chapovalov M., Lebedev A., Leites D., The classification of almost affine (hyperbolic) Lie superalgebras, J. Nonlinear Math. Phys. 17 (2010), suppl. 1, 103-161, arXiv:0906.1860.
-
Deligne P., Morgan J.W., Notes on supersymmetry (following Joseph Bernstein), in Quantum Fields and Strings: a Course for Mathematicians, Vols. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, 41-97.
-
Dooley A.H., Rice J.W., On contractions of semisimple Lie groups, Trans. Amer. Math. Soc. 289 (1985), 185-202.
-
Fioresi R., Gavarini F., Chevalley supergroups, Mem. Amer. Math. Soc. 215 (2012), vi+64 pages, arXiv:0808.0785.
-
Gavarini F., Chevalley supergroups of type $D(2,1;a)$, Proc. Edinb. Math. Soc. 57 (2014), 465-491, arXiv:1006.0464.
-
Gavarini F., Global splittings and super Harish-Chandra pairs for affine supergroups, Trans. Amer. Math. Soc. 368 (2016), 3973-4026, arXiv:1308.0462.
-
Gavarini F., Lie supergroups vs. super Harish-Chandra pairs: a new equivalence, arXiv:1609.02844.
-
Iohara K., Koga Y., Central extensions of Lie superalgebras, Comment. Math. Helv. 76 (2001), 110-154.
-
Kac V.G., Classification of simple algebraic supergroups, Russian Math. Surveys 32 (1977), no. 3, 214-215, in Russian, available at http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=3212&option_lang=eng.
-
Kac V.G., Lie superalgebras, Adv. Math. 26 (1977), 8-96.
-
Kaplansky I., Graded Lie algebras I, University of Chicago Report, 1976, available at http://www1.osu.cz/~zusmanovich/links/files/kaplansky/.
-
Kaplansky I., Graded Lie algebras II, University of Chicago Report, 1976, available at http://www1.osu.cz/~zusmanovich/links/files/kaplansky/.
-
Scheunert M., The theory of Lie superalgebras. An introduction, Lecture Notes in Math., Vol. 716, Springer, Berlin, 1979.
-
Serganova V., On generalizations of root systems, Comm. Algebra 24 (1996), 4281-4299.
-
Vaintrob A.Yu., Deformation of complex superspaces and coherent sheaves on them, J. Sov. Math. 51 (1990), 2140-2188.
-
Veisfeiler B.Ju., Kac V.G., Exponentials in Lie algebras of characteristic $p$, Math. USSR Izv. 5 (1971), 777-803.
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