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

Abstract

The complex Lie superalgebras g of type D(2, 1; a) - also denoted by 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 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 S3-symmetry of the whole situation, we shall work (following Kaplansky) with a two-dimensional parameter σ=(σ₁,σ₂,σ₃) ranging in the complex affine plane σ₁+σ₂+σ₃=0.