Représentation coadjointe quotient et espaces homogènes de contact ou localement conformément symplectiques. (Quotient coadjoint representation and homogeneous contact or locally conformally symplectic spaces) (Q1085814)

The starting point of this paper is the dual space \({\mathcal G}^*\) of a real Lie algebra \({\mathcal G}\) of dimension n. \({\mathcal G}^*\) is equipped with a natural Poisson structure. A sphere \(S^{n-1}\) with a natural Jacobi structure is obtained as a quotient by the group of positive homotheties. The leaves of \(S^{n-1}\) are the orbits of the quotient coadjoint mapping. The notion of quotient coadjoint representation of a Lie algebra or of a convex Lie group is introduced. A systematic study of this representation leads to the notion of homogeneous contact spaces (of odd dimension) and of locally conformally symplectic homogeneous spaces (of even dimension). A homogeneous contact space is a covering of a leaf of \(S^{n-1}\), which in turn may be proper or improper. An improper homogeneous contact space admits a corresponding Pfaffian homogeneous space as a covering. The results for locally conformally symplectic homogeneous spaces are similar.