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1 -cocycles on the group of contactomorphisms on the supercircle S 1 | 3 generalizing the Schwarzian derivative

Boujemaa Agrebaoui, Raja Hattab (2016)

Czechoslovak Mathematical Journal

The relative cohomology H diff 1 ( 𝕂 ( 1 | 3 ) , 𝔬𝔰𝔭 ( 2 , 3 ) ; 𝒟 λ , μ ( S 1 | 3 ) ) of the contact Lie superalgebra 𝕂 ( 1 | 3 ) with coefficients in the space of differential operators 𝒟 λ , μ ( S 1 | 3 ) acting on tensor densities on S 1 | 3 , is calculated in N. Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1 -cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative 1 -cocycle s ( X f ) = D 1 D 2 D 3 ( f ) α 3 1 / 2 , X f 𝕂 ( 1 | 3 ) which is invariant with respect to the conformal subsuperalgebra 𝔬𝔰𝔭 ( 2 , 3 ) of 𝕂 ( 1 | 3 ) . In this work we study the supergroup case. We give an explicit construction of 1 -cocycles of the group...

A rigidity phenomenon for germs of actions of R 2

Aubin Arroyo, Adolfo Guillot (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

We study germs of Lie algebras generated by two commuting vector fields in manifolds that are maximal in the sense of Palais (those which do not present any evident obstruction to be the local model of an action of  R 2 ). We study three particular pairs of homogeneous quadratic commuting vector fields (in  R 2 , R 3 and  R 4 ) and study the maximal Lie algebras generated by commuting vector fields whose 2-jets at the origin are the given homogeneous ones. In the first case we prove that the quadratic algebra...

Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras

Johannes Huebschmann (2000)

Banach Center Publications

Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms of certain associated differential (bi)graded Lie and G(erstenhaber)-algebras; in particular the G-algebra arising from an almost complex structure is a (strict) d(ifferential) G-algebra...

Dual vector fields ii: calculating the Jacobian

Philip Feinsilver, René Schott (2006)

Banach Center Publications

Given a Lie algebra with a chosen basis, the change of coordinates relating coordinates of the first and second kinds near the identity of the corresponding local group yields some remarkable vector fields and dual vector fields. One family of vector fields is dual to a representation of the Lie algebra acting on a Fock-type space. To this representation an abelian family of dual vector fields is associated. The exponential of these commuting operators acting on an appropriate vacuum yields the...

Equivalence of control systems with linear systems on Lie groups and homogeneous spaces

Philippe Jouan (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only if the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists,...

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