Homogeneous supermanifolds associated with the complex projective line
type: Item
instance of: scholarly article
title: Homogeneous supermanifolds associated with the complex projective line
zbMATH Open document ID: 0887.58003
DOI: 10.1007/BF02363912
author: V. A. Bunegina
author: A. L. Onishchik
published in: Journal of Mathematical Sciences (New York)
publication date: 1998-05-25T00:00:00Z
review text: Any Lie algebra of holomorphic vector fields defined on an open set of \(\mathbb{C}\) can be globalized to the Lie algebra of fundamental vector fields corresponding to the standard action of \(SL_2 (\mathbb{C})\) on \(\mathbb{C} P^1\), to the standard action of the affine group on \(\mathbb{C} \), or to the standard action of \(\mathbb{C}\) on itself. The authors consider a similar problem for Lie superalgebras of holomorphic vectors fields on the superspace \(\mathbb{C}^{1|m}\), where \(m>0\). The goal of the paper is to take a first step in this direction. Homogeneous supermanifolds (in the sense of Berezin-Leites) \((M, {\mathcal O})\) of dimension \(1\mid m\) such that \(M= \mathbb{C} P^1\) are studied. In the split case such supermanifolds are in a bijective correspondence with the nonincreasing \(m\)-tuples of nonnegative integers. The classification in the nonsplit case is not as simple and reduces to calculations with the tangent sheaf cohomology of split homogeneous supermanifolds. In the dimension \(1|2\) there is only one nonsplit homogeneous supermanifold of the form \((\mathbb{C} P^1, {\mathcal O})\): it is the superquadric in \(\mathbb{C} P^{1|2}\). When \(m=3\), there is a series of nonsplit homogeneous supermanifolds. The classification of such supermanifolds when \(m\geq 4\) is not yet completed.
reviewed by: A. P. Stone
Mathematics Subject Classification ID: 58A50
Mathematics Subject Classification ID: 58C50
zbMATH DE Number: 1017573
zbMATH Keywords: complex projective line
zbMATH Keywords: superspace
zbMATH Keywords: superalgebra
zbMATH Keywords: holomorphic vectors fields
zbMATH Keywords: homogeneous supermanifolds
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1017573