Singular Degenerations of Lie Supergroups of Type $D(2,1;a)$

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 $\mathbf{\sigma }=({\sigma }_1,{\sigma }_2,{\sigma }_3)$ ranging in the complex affine plane ${\sigma }_1+{\sigma }_2+{\sigma }_3=0$.

References in corpus (2)

• Classification of Finite Dimensional Modular Lie Superalgebras with Indecomposable Cartan Matrix
• Lie supergroups vs. super Harish-Chandra pairs: a new equivalence