The Lie superalgebra of a supermanifold. (English) Zbl 1207.58005
Summary: 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.
MSC:
58A50 | Supermanifolds and graded manifolds |
17B66 | Lie algebras of vector fields and related (super) algebras |
14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
17B70 | Graded Lie (super)algebras |
17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |