To the Homogeneous Symplectic Manifold toward the Geometry of information

Abstract

Every symplectic Lie group (G,ω) carries local flat left- invariant structure $(G,\nabla)$ given by the formula $\omega(\nabla(X,Y),Z) = \omega(\mu(X,Z),Y)$ where X,Y and Z are elements of $\mathfrak{g} = (V,\mu)$ the Lie algebra deriving from G. There exists a KV-complex (C,d) on $(G,\nabla)$ of which we give the direct factor H ²(G,ℝ) of H ²(C,d). This factor permit us, with other supplementary information, to classify the Lie algebras of dimension ≤ 6 which are Lefschetz coming from symplectic Lie groups spanning them. The study of these homogeneuos manifolds give some useful results to the geometry of information.