Algebroids associated with pre-Poisson structures

Abstract

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.