Journal of Lie Theory 23 (2013), No. 1, 035--054
Copyright Heldermann Verlag 2013
Lie Superalgebras of Differential Operators
Janusz Grabowski
Polish Academy of Sciences, Institute of Mathematics, Sniadeckich 8 -- P.O. Box 21, 00-956 Warsaw, Poland
jagrab@impan.pl
Alexei Kotov
Faculté des Sciences, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg City, Luxembourg
and: Institute of Mathematics and Statistics, University of Tromso, 9037 Tromso, Norway
oleksii.kotov@uit.no
Norbert Poncin
Faculté des Sciences, University of Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359 Luxembourg City, Luxembourg
norbert.poncin@uni.lu
We describe explicitly Lie superalgebra isomorphisms between the Lie superalgebras of first-order superdifferential operators on supermanifolds, showing in particular that any such isomorphism induces a diffeomorphism of the supermanifolds. We also prove that the group of automorphisms of such a Lie superalgebra is a semi-direct product of the subgroup of automorphisms induced by the supermanifold diffeomorphisms and another subgroup which consists of automorphisms determined by even superdivergences. We prove the existence of such superdivergences on any supermanifold and we describe their local form.
Keywords: Supermanifold, Lie superalgebra, differential operators, vector fields, automorphisms, Lie superalgebra cohomology, divergence.
MSC: 58A50, 17B40, 17B66, 13N10, 17B56
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