Cohomology of Koszul-Vinberg algebroids and Poisson manifolds. I (Q2782567)

This paper gives an account of some results on Koszul-Vinberg algebroids and their cohomology theory. A (not necessarily associative) \(\mathbb{R}\)-algebra \(A \) is called a Koszul-Vinberg algebra (or left symmetric algebra) if \((a,b,c) =(b,a,c) \) for all \(a,b,c\in A\) where \((a,b,c) \) represents the ``associator'' product \((a,b,c) =a(bc) -(ab) c\). A non-associative \(A\)-bimodule \(W\), i.e. an \(\mathbb{R}\)-vector space \(W\) endowed with bilinear maps \(A\times W\to W\) and \(W\times A\to W\), is called a Koszul-Vinberg module of \(A\) if \((a,b,w) =(b,a,w) \) and \((a,w,b) =(w,a,b) \) for all \(a,b\in A\) and \(w\in W\), where the triples are ``associator'' products defined in a similar way as above. There is a cohomology theory for the pair \((A,W) \) of a Koszul-Vinberg algebra \(A\) and its module \(M\) built from cochain spaces \( C^{q}(A,W) \) of \(q\)-linear maps from \(A\) to \(W\). Furthermore, a Koszul-Vinberg algebroid over a manifold \(M\) is a vector bundle \(E\) over \(M\) with a bundle map \(a:E\to TM\), called the anchor map, such that the space \(\Gamma (E) \) of smooth cross sections of \(E\) is a Koszul-Vinberg algebra with \((fs) s'=f(ss') \) and \(s(fs') =f(ss') +\langle df,a(s) \rangle s'\) for all \(f\in C^{\infty}(M,\mathbb{R}) \) and \(s,s'\in \Gamma (E) \). Note that the space \(C^{\infty}(M,\mathbb{R}) \) is a Koszul-Vinberg module of \(\Gamma (E) \) with the module structure defined by \(sf=\langle df,a(s) \rangle \) and \(fs=0\) for \(s\in \Gamma (E) \) and \(f\in C^{\infty}(M,\mathbb{R}) \). The tangent bundle \(TM\) of a locally flat manifold \((M,D) \) is an example of a Koszul-Vinberg algebroid over \(M\) with the identity map of \(TM\) as the anchor map, where the space \(\Gamma (TM) \) of smooth vector fields on \(M\) is a Koszul-Vinberg algebra with product defined by \(XY=D_{X}Y\) for \(X,Y\in \Gamma (TM) \). The author describes some relations between the cohomology for the pair \((\Gamma (\mathcal{E}) ,C^{\infty}(M,\mathbb{R})) \) and the Poisson structures on \(M\), where \(\mathcal{E}\) is a Koszul-Vinberg algebroid \(E\to M\) augmented by the trivial bundle \(M\times \mathbb{R}\) over \(M\). Furthermore, the author considers the problem of deforming the Koszul-Vinberg algebra structure on \(\Gamma (\mathcal{E}) \) via ``star products'' \(\ast _{h}\) on \(\Gamma (\mathcal{E}) \) such that each \((\Gamma (\mathcal{E}) ,\ast _{h}) \) is still a Koszul-Vinberg algebra, where \(\xi _{1}\ast _{h}\xi _{2}\) is of the form \(\xi _{1}\xi _{2}+\sum_{k>0}h^{k}\theta _{k}(f_{1},f_{2}) \) with \(\theta _{k}\in C^{2}(\Gamma (\mathcal{E}) ,\Gamma (\mathcal{E})) \) and \(\xi _{1}\xi _{2}\) is the original product in \(\Gamma (\mathcal{E}) \) for \(\xi _{i}=(s_{i},f_{i}) \in \Gamma (\mathcal{E}) \) with \(s_{i}\in \Gamma (E) \) and \(f_{i}\in C^{\infty}(M,\mathbb{R}) \).NEWLINENEWLINEFor the entire collection see [Zbl 0980.00030].