Kentaro Mikami

Algebroids associated with pre-Poisson structures

There are several ways to generalize Poisson structures. A Jacobi structure (or a local Lie algebra structure), in which we do not require the Leibniz identity for the bracket, and a Nambu-Poisson structure, where the brackets are not binary but n-ary operations satisfying a generalized Leibniz rule called fundamental identity, are well-known examples. Also, a Dirac structure is a natural generalization of a Poisson structure. As another direction of studying Poisson geometry, we would like to do some trial or attempt to generalize the concepts, ideas, or theories of Poisson geometry into some area where the Poisson condition is not fulfilled. In the first half of this note, we show briefly our trials in this context, namely in almost Poisson geometry. As we will see in short, a Poisson structure gives a Lie algebroid. It is natural to handle a Leibniz algebroid as generalization of a Lie algebroid. Thus, it is meaningful to study the fundamental properties of Leibniz algebra or super Leibniz algebra. In the second half of this note, after we recall some properties of Leibniz modules, we define super Leibniz algebras and super Leibniz modules keeping the exterior algebra bundle of the tangent bundle with Schouten bracket as a prototype of a super Lie algebra (and so a super Leibniz algebra). We will show that an abelian extension is controlled by the second super cohomology group. The notion of super Leibniz bundles is clear, but unfortunately we do not have the proper notion of anchor, so far. In near future, we hope we could find concrete examples of super Leibniz bundles tightly connected to the properties of Poisson geometry, and could understand what the anchor should be.

INTEGRABILITY OF PLANE FIELDS DEFINED BY 2-VECTOR FIELDS

Given a 2-vector field on a manifold, first we briefly discuss the complete integrability of the distribution which is the image of the 2-vector field. Then we show that a new Lie algebroid is defined on such a maniold which is coincident with the cotangent Lie algebroid when the 2-vector field is Poisson. The result is extended to the case of Lie algebroids.

Lie Algebroids Associated with Deformed Schouten Bracket of 2-Vector Fields

Given a 2-vector field and a closed 1-form on a manifold, we consider the set of cotangent vectors which annihilate the deformed Schouten bracket of the 2-vector field by the closed 1-form. We show that if the space of cotangent vectors forms a vector bundle, it carries a structure of a Lie algebroid. We treat this theorem in the category of Lie algebroids. As a special case, this result contains the well-known fact that the 1-jet bundle of functions of a contact manifold has a Lie algebroid structure.

An Interpretation of the Schouten-Nijenhuis Bracket

The Schouten-Nijenhuis bracket ([4],[5]) is used to describe whether a 2-vector field becomes a Poisson tensor field or not, due to Lichnerowicz, and it is a very popular and useful tool in Poisson geometry. In this note we observe that the bracket has a very natural and simple notion in its root, namely, skew-symmetry and the Leibniz’ rule. We then give an easy proof for the super Jacobi identity of the Schouten-Nijenhuis bracket.