Journal of Lie Theory 20 (2010), No. 4, 739--749
Copyright Heldermann Verlag 2010
The Lie Superalgebra of a Supermanifold
Janusz Grabowski
Polish Academy of Sciences, Institute of Mathematics, ul. Sniadeckich 8, 00-956 Warsaw, Poland
jagrab@impan.pl
Alexei Kotov
Mathematics Research Unit, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg City, Luxembourg
alexei.kotov@uni.lu
Norbert Poncin
Mathematics Research Unit, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg City, Luxembourg
norbert.poncin@uni.lu
We prove a "superversion" of Shanks and Pursell's classical result stating that any isomorphism of the Lie algebras of compactly supported vector fields is implemented by a diffeomorphism of underlying manifolds. We thus provide a Lie algebraic characterization of supermanifolds and describe explicitly isomorphisms of the Lie superalgebras of supervector fields on supermanifolds.
Keywords: Superalgebra, noncommutative space, supermanifold, graded manifold, super vector field, graded Lie algebra.
MSC: 58A50, 17B66, 14F05, 17B70, 17B40
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