On isotropic super-Grassmannians of maximal type associated with an odd bilinear form

Abstract

We study the super-Grassmannian IGr n|n,s|t (C) formed by those graded subspaces of dimension s|t, s + t = n, of the vector superspace C n|n that are totally isotropic with respect to an odd skew-symmetric bilinear form b, defined in this superspace. We prove that IGr n|n,s|t (C) is a non-split supermanifold whenever t ≥ 1, s ≥ 2, and a rigid one whenever t ≥ 3, s ≥ 4. If t ≥ 2, s ≥ 3, then the Lie superalgebra of vector fields on this super-Grassmannian coincides with the simple classical Lie superalgebra πsp n|n (C) that annihilates the form b.

On complex lie supergroups and split homogeneous supermanifolds

“…In [16] it was proved that the isotropic super-Grassmannian of maximal type IGr n|n,s|t (C) associated with an odd bilinear form is non-split whenever t ≥ 1 and s ≥ 2. In [15] the complete solution of the problem was given for the isotropic super-Grassmannian of maximal type associated with an even bilinear form.…”

“…In [15] the complete solution of the problem was given for the isotropic super-Grassmannian of maximal type associated with an even bilinear form. Note that the method of [15] and [16] can be used for all series of flag supermanifolds.…”

Vector fields on osp2m|2n(C)- and Vishnyakova1 2018Journal of Algebra3010View full textAdd to dashboardCite

“…Manin defined four series of complex homogeneous supermanifolds that correspond to four series of linear Lie superalgebras: gl m|n (C), osp m|2n (C), πsp n (C) and q n (C). The Lie superalgebras of holomorphic vector fields on these supermanifolds were studied in [BO,Bun,O1,O2,OS1,OS2,OS3,Ser] in the case of super-Grassmannians and in [V1, V2, V4, V6] in the case of other flag supermanifolds. We summarize the obtained results in the following table : Super-Grassmannians Other flag supermanifolds gl m|n (C) + + (generic) osp 2m|2n (C) + (maximal type) + (maximal type) osp 2m+1|2n (C) + (maximal type) πsp n|n (C) + (maximal type) + (maximal type) q n|n (C) + +…”

On the splitting problem for complex homogeneous supermanifolds

“…The calculation of the Green moduli space is a difficult problem itself, and in many cases the method is difficult to apply. Furthermore, Onishchik and Serov [9,10,11] considered grading derivations, which correspond to Z-gradings of the structure sheaf of a supermanifold. For example, it was shown that almost all supergrassmannians do not possess such derivations, i.e.…”

“…For any h ∈ G 0 , denote by r ′ h and l ′ h the right and the left translation in the Lie supergroup G 3 = (G 0 , Hom C ( g1, F G 0 )), respectively. (See, (11)…”