Search results for "Lie bracket of vector fields"

showing 3 items of 3 documents

Existence and uniqueness of solutions to superdifferential equations

1993

Abstract 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 = X0 + X1, has a unique integral flow, Г: R 1¦1 x (M, AM) → (M, AM), satisfying a given initial condition. A necessary and sufficient condition for this integral flow to yield an R 1¦1-action is obtained: the homogeneous components, X0, and, X1, of the given field must define a Lie superalgebra of dimension (1, 1). The supergroup structure on R 1¦1, however, has to be specified: there are three non-isomorphic Lie supergroup structures on R 1¦1, all of which have addition as the group operation in the underlying …

Flow (mathematics)Simple Lie groupMathematical analysisLie bracket of vector fieldsAdjoint representationGeneral Physics and AstronomyLie groupLie derivativeLie superalgebraGeometry and TopologySupergroupMathematical PhysicsMathematicsJournal of Geometry and Physics
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The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

2010

Abstract In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V k to a decreasing family of k foliations F i on a manifold M . We have shown that there exists a ( 1 , 1 ) tensor J of V k such that J k ≠ 0 , J k + 1 = 0 and we defined by L J ( V k ) the Lie Algebra of vector fields X on V k such that, for each vector field Y on V k , [ X , J Y ] = J [ X , Y ] . In this note, we study the first Chevalley–Eilenberg Cohomology Group, i.e. the quotient space of derivations of L J ( V k ) by the subspace of inner derivations, denoted by H 1 ( L J ( V k ) ) .

Pure mathematicsFoliacions (Matemàtica)Group (mathematics)General Physics and AstronomyLie Àlgebres deQuotient space (linear algebra)CohomologyAlgebraTensor (intrinsic definition)Lie bracket of vector fieldsLie algebraVector fieldFiber bundleGeometry and TopologyMathematical PhysicsMathematicsJournal of Geometry and Physics
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Poisson-Nijenhuis structures and the Vinogradov bracket

1994

We express the compatibility conditions that a Poisson bivector and a Nijenhuis tensor must fulfil in order to be a Poisson-Nijenhuis structure by means of a graded Lie bracket. This bracket is a generalization of Schouten and Frolicher-Nijenhuis graded Lie brackets defined on multivector fields and on vector valued differential forms respectively.

Schouten–Nijenhuis bracketGraded Lie algebraAlgebraFrölicher–Nijenhuis bracketPoisson bracketAdjoint representation of a Lie algebraNonlinear Sciences::Exactly Solvable and Integrable SystemsMathematics::Quantum AlgebraPoisson manifoldLie bracket of vector fieldsLie derivativeMathematics::Differential GeometryGeometry and TopologyMathematics::Symplectic GeometryAnalysisMathematicsAnnals of Global Analysis and Geometry
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