SINGULAR DEGENERATIONS OF LIE SUPERGROUPS OF TYPE D(2, 1; a)

International audienceThe complex Lie superalgebras g of type D(2, 1; a) are usually defined 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 those singular values too, giving rise to " singular specializations " that are no longer simple. This extends (in five different ways) the classically known D(2, 1; a) family. Basing on this construction, we perform the parallel one for complex Lie supergroups and describe their singular specializations (or " degenerations ") at singular values of the parameter. This is done via a general construction based on suitably chosen super Harish-Chandra pairs, which suits the Lie group theoretical framework; nevertheless, it might also be realized by means of a straightforward extension of the method introduced in [FG] and [Ga1] to construct " Chevalley supergroups " , which is fit for the context of algebraic supergeometry. Although one may adopt Kac' presentation for the Lie superalgebras of type D(2, 1; a) , in order to stress the overall S 3 –symmetry of the whole situation, we shall work (like Scheunert does, for instance, see [Sc]) with a two-dimensional parameter σ = (σ 1 , σ 2 , σ 3) ranging in the complex affine plane σ 1 + σ 2 + σ 3 = 0 instead of the single parameter a ∈ C