Globalizations of infinitesimal actions on supermanifolds (Q2927940)

An infinitesimal action of a Lie superalgebra \(\mathfrak{g}\) on a supermanifold \(\mathcal{M}\) is a Lie superalgebras homomorphism from \(\mathfrak{g}\) to the Lie superalgebra of global vector fields on \(\mathcal{M}\). A local action of a Lie supergroup \(\mathcal{G}\) on a supermanifold \(\mathcal{M}\) is a morphism from an open submanifold of \(\mathcal{G}\times \mathcal{M}\) to \(\mathcal{M}\) satisfying natural conditions.NEWLINENEWLINEThe relations of infinitesimal, local and global actions on smooth manifolds were studied by \textit{R. S. Palais} [Mem. Am. Math. Soc. 22, 123 p. (1957; Zbl 0178.26502)]. In particular, a globalization of a local action of a Lie group \(G\) on a smooth manifold \(M\) is an extension of this action to a global action of \(G\) on a smooth (non-Hausdorff) manifold \(M'\) containing \(M\) as an open submanifold and such that \(G\cdot M=M'\).NEWLINENEWLINEIn the present paper, the author shows that an infinitesimal action of a Lie superalgebra and a local action of the corresponding Lie supergroup on a supermanifold define each other. Next, conditions for a local action of Lie supergroups to be globalized are found (the notion of globalization in the super settings naturally generalizes the usual one). As a corollary it is shown that an infinitesimal action of a Lie superalgebra on a supermanifold with the underling compact smooth manifold can be extended to a global action of the corresponding simply connected Lie supergroup on the same supermanifold.NEWLINENEWLINEThe author uses some ideas of Palais, but the theory of supermanifolds contains many phenomenons requiring many additional considerations. The paper is written clearly and all necessary notions and facts used in the proofs are explained.