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

SIGMA 14 (2018), 137, 36 pages      arXiv:1709.04717

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

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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