### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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

1. 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.
2. 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.
3. 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.
4. 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.
5. Dooley A.H., Rice J.W., On contractions of semisimple Lie groups, Trans. Amer. Math. Soc. 289 (1985), 185-202.
6. Fioresi R., Gavarini F., Chevalley supergroups, Mem. Amer. Math. Soc. 215 (2012), vi+64 pages, arXiv:0808.0785.
7. Gavarini F., Chevalley supergroups of type $D(2,1;a)$, Proc. Edinb. Math. Soc. 57 (2014), 465-491, arXiv:1006.0464.
8. Gavarini F., Global splittings and super Harish-Chandra pairs for affine supergroups, Trans. Amer. Math. Soc. 368 (2016), 3973-4026, arXiv:1308.0462.
9. Gavarini F., Lie supergroups vs. super Harish-Chandra pairs: a new equivalence, arXiv:1609.02844.
10. Iohara K., Koga Y., Central extensions of Lie superalgebras, Comment. Math. Helv. 76 (2001), 110-154.
11. 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.
12. Kac V.G., Lie superalgebras, Adv. Math. 26 (1977), 8-96.
13. Kaplansky I., Graded Lie algebras I, University of Chicago Report, 1976, available at http://www1.osu.cz/~zusmanovich/links/files/kaplansky/.
14. Kaplansky I., Graded Lie algebras II, University of Chicago Report, 1976, available at http://www1.osu.cz/~zusmanovich/links/files/kaplansky/.
15. Scheunert M., The theory of Lie superalgebras. An introduction, Lecture Notes in Math., Vol. 716, Springer, Berlin, 1979.
16. Serganova V., On generalizations of root systems, Comm. Algebra 24 (1996), 4281-4299.
17. Vaintrob A.Yu., Deformation of complex superspaces and coherent sheaves on them, J. Sov. Math. 51 (1990), 2140-2188.
18. Veisfeiler B.Ju., Kac V.G., Exponentials in Lie algebras of characteristic $p$, Math. USSR Izv. 5 (1971), 777-803.