Triangular generalized Lie bialgebroids: Homology and cohomology theories (Q2782568)

\textit{K. C. H. Mackenzie} and \textit{P. Xu} [Duke Math. J. 73, No. 2, 415-452 (1994; Zbl 0844.22005)] introduced Lie bialgebroids as the infinitesimal invariants of Poisson groupoids. In their sequential papers, they showed that there is an associated Poisson groupoid structure for the Lie bialgebroid of a simply connected Lie groupoid. If \(M\) is a Poisson manifold, it induces a Lie algebroid structures on \(TM\) and \(T^*M\), and the pair \((TM, T^*M)\) is an example of a Lie bialgebroid. A Jacobi manifold is a triple \((M,\Lambda, E)\), where \(M\) is a manifold, \(\Lambda\) a 2-vector and \(E\) a vector field satisfying \([\Lambda,\Lambda] = 2 E\wedge\Lambda\) and \([E,\Lambda] = 0\), where \([\;,\;]\) denotes the Shouten-Nijenhuis bracket. Jacobi manifolds are natural generalizations of Poisson and contact manifolds. The 1-jet bundle \(T^*M\times R\to M\) of the Jacobi manifold \(M\) admits a Lie algebroid structure. But \((TM\times R, T^*M\times R)\) is not a Lie bialgebroid if \(TM\times R\to M\) carries the natural Lie algebroid structure.NEWLINENEWLINENEWLINEFinding a natural generalization is the main motivation for the paper under review. It studies the homology and cohomology associated with the dual Lie algebroid \(A^*\) of a triangular generalized Lie bialgebroid \((A,\varphi_0, p)\), proves that the vanishing property of a certain cohomology class implies the duality between these homology and cohomology theories.NEWLINENEWLINENEWLINERecall that \((A,\varphi_0, p)\) is a triangular generalized Lie bialgebroid if \(\varphi_0\) is a 1-cycle in the Lie algebroid cohomology complex of \(A\) with trivial coefficients, \(p\in\Gamma(\wedge^2 A)\) is a section of \(\wedge^2 A\to M\) such that \([p, p] = 2 i(\varphi_0) p\wedge p\), \(X_0 = - i(\varphi_0)p\) is a 1-cocycle of \(A^*\), both \(A\) and \(A^*\) are Lie algebroids and \(\varphi_0, X_0\) satisfy some compatibility conditions. In the case \(\varphi_0 = 0\) and \(X_0 = 0\) (or \(p =0\)) one recovers the Lie bialgebroids defined by Mackenzie and Xu.NEWLINENEWLINENEWLINEIn section 2, Lie algebroids, Jacobi manifolds, cohomology of Lie algebras are briefly recalled. The homology of Lie algebroids with respect to a flat connection is determined. \(\varphi_0\)-cohomology of Lie algebroids and the \(\varphi_0\)-Schouten bracket are discussed in section 3. The differential of the cohomology of a triangular generalized Lie bialgebroid is determined in Proposition 4.2, similarly the operator for the homology of a triangular generalized Lie bialgebroid is studied in Theorem 4.8. The main result on the duality between homology and cohomology of unimodular triangular generalized bialgebroid \((A,\varphi_0,p)\) is given in Theorem 4.10. The paper recovers some previous results for unimodular Poisson and Jacobi manifolds and unimodular triangular Lie bialgebroids. Many examples are given.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00030].