Existence and Uniqueness of Solution to Superdifferential Equations
We state and.prove the theorem of existence and uniqueness of solutions to ordinary superdifferential equations on supermanifolds. It is shown that any supei'vector. field, X= Xo+Xt, ii~ a unique integral flow, r.: JR111 X (M,AM)--+ (M,AM), satisfying a given initial condition. A necessary and...
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| Tipo de recurso: | informe técnico |
| Estado: | Versión publicada |
| Fecha de publicación: | 1992 |
| País: | México |
| Institución: | Centro de Investigación en Matemáticas |
| Repositorio: | Repositorio Institucional CIMAT |
| Idioma: | inglés |
| OAI Identifier: | oai:cimat.repositorioinstitucional.mx:1008/785 |
| Acceso en línea: | http://cimat.repositorioinstitucional.mx/jspui/handle/1008/785 |
| Access Level: | acceso abierto |
| Palabra clave: | info:eu-repo/classification/MSC/Ecuaciones Supediferenciales info:eu-repo/classification/cti/1 info:eu-repo/classification/cti/12 info:eu-repo/classification/cti/1299 info:eu-repo/classification/cti/129999 |
| Sumario: | We state and.prove the theorem of existence and uniqueness of solutions to ordinary superdifferential equations on supermanifolds. It is shown that any supei'vector. field, X= Xo+Xt, ii~ a unique integral flow, r.: JR111 X (M,AM)--+ (M,AM), satisfying a given initial condition. A necessary and sufficient condition for this integral flow to yield an R111-action is obtained: the homogeneous components, Xo, and, X1, of the given field must define a Lie superalgebra of dimension (1, 1). The supergroup structure on JR111 however, has to be specified: there are three non-isomorphic Lie supergroup structures on R111 , all of which have addition as the group operation in the underlying Lie group R. On the other extreme, even if Xo, and X1 do not close to form a Lie superalgebra, the integral flow of X is uniquely determined arid is independent of the Lie ·supergroup structure imposed on JR111 . This fact makes it possible to establish an unambiguous relationship between the algebraic Lie deri'vative of supergeometric obJ~cts (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 R111 is fixed, some flows obtained from left-invariant supervector fields on Lie supergroups may fail to define an R111-action of the choa~n struc.ture. Finally, necessary and sufficient conditions for the integral flows of two supervector fields to commute are given. |
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