Poisson structures on almost complex Lie algebroids

Abstract

In this paper we extend the almost complex Poisson structures from almost complex manifolds to almost complex Lie algebroids. Examples of such structures are also given and the almost complex Poisson morphisms of almost complex Lie algebroids are studied.

Almost Lie Algebroids and Characteristic Classes

“…Analogously to the Lie algebroid case [11,19], an almost complex almost Lie algebroid is a real almost Lie algebroid E such that there is an almost complex endomorphism on E (i.e., an endomorphism…”

“…The interest in E 0 supporting an almost Lie algebroid structure comes not only from the fact that it has no Lie algebroid structure, but also from the fact that it can not be a Courant vector bundle (Proposition 2.10). But according to the definitions related to the almost complex Lie algebroid case in [11,19], E 0 has an almost complex endomorphism that has a null Nijenhuis tensor, i.e., it is integrable (Proposition 2.11).…”

Laplace operators on holomorphic Lie algebroids

“…Real Lie algebroids have been studied by A. Weinstein [21], P. Popescu [17,18], M. Anastasiei [2], L. Popescu [20]. Complex and holomorphic Lie algebroids have been investigated by C.-M. Marle [11], P. Popescu [19], P. Popescu, C. Ida [6]. E. Martinez [12,13] has introduced the notion of prolongation of a Lie algebroid, as a tool for studying the geometry of a Lie algebroid in a context which is similar to the tangent bundle of a manifold.…”

“…Real Lie algebroids have been studied by A. Weinstein [21], P. Popescu [17,18], M. Anastasiei [2], L. Popescu [20]. Complex and holomorphic Lie algebroids have been investigated by C.-M. Marle [11], P. Popescu [19], P. Popescu, C. Ida [6].…”

On Almost Complex Lie Algebroids

“…We notice that the present paper can be considered as an introduction of basic elements of almost complex geometry in the almost complex Lie algebroids framework. Some of these notions are continued in [36] where almost complex Poisson structures on almost complex Lie algebroids are studied, but ohter problems related to almost complex geometry or complex (holomorphic) geometry as for instance: Laplacians or vanishing theorems are still open in the framework of Lie algebroids, as well as the study of anti-Hermitian (Kählerian) Lie algebroids. Also taking into account the role of almost complex geometry in the study of almost contact geometry the present notions can be useful in the study of almost contact or contact structures on Lie algebroids.…”