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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Remarks on Grassmannian supermanifolds

Author: Oscar Adolfo Sánchez-Valenzuela
Journal: Trans. Amer. Math. Soc. 307 (1988), 597-614
MSC: Primary 58A50; Secondary 14M15
MathSciNet review: 940219
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Abstract: This paper studies some aspects of a particular class of examples of supermanifolds; the supergrassmannians, introduced in [Manin]. Their definition, in terms of local data and glueing isomorphisms, is reviewed. Explicit formulas in local coordinates are given for the Lie group action they come equipped with. It is proved that, for those supergrassmannians whose underlying manifold is an ordinary grassmannian, their structural sheaf can be realized as the sheaf of sections of the exterior algebra bundle of some canonical vector bundle. This realization holds true equivariantly for the Lie group action in question, thus making natural in these cases the identification given in [Batchelor]. The proof depends on the computation of the transition functions of the supercotangent bundle as defined in a previous work [OASV 2]. Finally, it is shown that there is a natural supergroup action involved (in the sense of [OASV 3]) and hence, the supergrassmannians may be regarded as examples of superhomogeneous spaces--a notion first introduced in [Kostant]. The corresponding Lie superalgebra action can be realized as superderivations of the structural sheaf; explicit formulas are included for those supergrassmannians identifiable with exterior algebra vector bundles.

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Article copyright: © Copyright 1988 American Mathematical Society

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