Kv-Cohomology of Koszul–vinberg Algebroids and Poisson Manifolds

Abstract

The main concern of this paper is the study of the relationships between the KV-cohomology of Koszul–Vinberg algebras and some properties of various geometrical objects. In particular we show how the scalar KV-cohomology of real or holomorphic Koszul–Vinberg algebroids is closely related to real or holomorphic Poisson manifolds. In the appendix we point out strong relationships between the pioneer work of Nijenhuis [42] and the KV-cohomology.

Integration of holomorphic Lie algebroids

“…Hence the Lie algebroid of Γ ⇒ X is isomorphic to (T * X) π R . By Proposition 2.6, (T * X) π R is the underlying real Lie algebroid of the holomorphic Lie algebroid (T * X) 1 4 π . According to Theorem 3.17, Γ ⇒ X admits a multiplicative almost complex structure J Γ which makes Γ into a holomorphic Lie groupoid.…”

“…Let (X, π), where π = π R + iπ I ∈ Γ(∧ 2 T 1,0 X), be a holomorphic Poisson manifold. Assume that Γ ⇒ X is a holomorphic Lie groupoid with almost complex structure J Γ , whose corresponding holomorphic Lie algebroid is (T * X) 1 4 π . Moreover, assume that there exists a symplectic real 2-form ω R on the underlying real Lie groupoid such that (Γ ⇒ X, ω R ) is a symplectic groupoid integrating the real Poisson structure π R .…”

“…Moreover, assume that there exists a symplectic real 2-form ω R on the underlying real Lie groupoid such that (Γ ⇒ X, ω R ) is a symplectic groupoid integrating the real Poisson structure π R . Then ω = ω R − iω I ∈ Ω 2,0 (Γ), where ω I (·, ·) = ω(J·, ·), is a multiplicative holomorphic symplectic 2-form on Γ so that (Γ ⇒ X, 1 4 ω) is a holomorphic symplectic groupoid integrating the holomorphic Poisson structure π.…”

Some lagrangian invariants of symplectic manifolds

“…KV-cohomology. In this subsection we recall the definition of the cohomology complex of KV-algebras with coefficients in their two-sided modules, see [NB3], [NB4]. Let V be a two-sided module over a KV-algebra A.…”

“…Given a regular foliation F let T F ⊂ T M be the tangent bundle of F . Then F is an affine foliation if and only if T F is a Koszul-Vinberg algebroid whose anchor map is the inclusion map, [NB4], [NB5], [NBW1], [NBW2]. Here are some interesting examples of affine foliations.…”