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Existence and uniqueness of solutions to superdifferential equations. (English) Zbl 0772.58007
Summary: We state and prove the theorem of existence and uniqueness of solutions to ordinary superdifferential equations on supermanifolds. It is shown that any supervector field \(X = X_ 0 + X_ 1\), has a unique integral flow, \(\Gamma: \mathbb{R}^{1| 1} \times (M,{\mathcal A}_ M) \to (M,\mathcal A_ M)\), satisfying a given initial condition. A necessary and sufficient condition for this integral flow to yield an \(\mathbb{R}^{1| 1}\)-action is obtained: the homogeneous components, \(X_ 0\), and \(X_ 1\), of the given field must define a Lie superalgebra of dimension (1,1). The supergroup structure on \(\mathbb{R}^{1| 1}\), however, has to be specified: there are three non-isomorphic Lie supergroup structures on \(\mathbb{R}^{1| 1}\), all of which have addition as the group operation in the underlying Lie group \(\mathbb{R}\). On the other extreme, even if \(X_ 0\), and \(X_ 1\) do not close to form a Lie superalgebra, the integral flow of \(X\) is uniquely determined and is independent of the Lie supergroup structure imposed on \(\mathbb{R}^{1| 1}\). This fact makes it possible to establish an unambiguous relationship between the algebraic Lie derivative of supergeometric objects (e.g., superforms), and its geometrical definition in terms of integral flows. It is shown by means of examples that if a supergroup structure in \(\mathbb{R}^{1| 1}\) is fixed, some flows obtained from left-invariant supervector fields on Lie supergroups may fail to define an \(\mathbb{R}^{1| 1}\)-action of the chosen structure. Finally, necessary and sufficient conditions for the integral flows of two supervector fields to commute are given.

MSC:
58A50 Supermanifolds and graded manifolds
58C50 Analysis on supermanifolds or graded manifolds
37C10 Dynamics induced by flows and semiflows
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A26 Geometric methods in ordinary differential equations
22E20 General properties and structure of other Lie groups
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