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Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$
- Source :
- SIGMA 14 (2018), 137, 36 pages
- Publication Year :
- 2017
-
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 $\sigma_1 + \sigma_2 + \sigma_3 = 0$.
- Subjects :
- Mathematics - Rings and Algebras
Mathematics - Differential Geometry
Subjects
Details
- Database :
- arXiv
- Journal :
- SIGMA 14 (2018), 137, 36 pages
- Publication Type :
- Report
- Accession number :
- edsarx.1709.04717
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.3842/SIGMA.2018.137