Jacobi Structures on Affine Bundles

Abstract

We study affine Jacobi structures (brackets) on an affine bundle π : A → M , i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non-zero, there is a one-to-one correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A + = p∈M Af f (Ap, R) of affine functionals. In the case rank A = 0, it is shown that there is a one-to-one correspondence between affine Jacobi structures on A and local Lie algebras on A + . Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited (strongly-affine or affine-homogeneous Jacobi structures) over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real finitedimensional Lie algebra.Mathematics Subject Classification (2000): 53D17, 53D05, 81S10.

Kirillov structures and reduction of Hamiltonian systems by scaling and standard symmetries

“…In particular, those in which the base space of the line bundle is the projective bundle associated with a vector bundle (for more details on Kirillov structures, see for instance, Refs. [31][32][33][34][35].…”

“…In this section, we recall some notions and properties of contact, Jacobi and Kirillov manifolds (for more details see, for instance, Refs. 3,29,[31][32][33][34][35][42][43][44].…”

“…In the sequel, we will describe this structure. The expression of the reduced Poisson structure on 𝑃 in terms of the new local coordinates (𝜌, 𝜌 ′ , 𝜎) is (see (33))…”