Hirzebruch signature operator for transitive Lie algebroids

Abstract

The aim of the paper is to construct Hirzebruch signature operator for transitive invariantly oriented Lie algebroids 1 Signature of Lie algebroids 1.1 De nition of Lie algebroids, Atiyah sequence Lie algebroids appeared as in nitesimal objects of Lie groupoids, principal bre bundles, vector bundles (Pradines, 1967), TC-foliations and nonclosed Lie subgroups (Molino, 1977), Poisson manifolds (Dazord, Coste, Weinstein, 1987), etc. Their algebraic equivalents are known as Lie pseudo-algebras (Herz 1953) called also further as Lie-Rinehart algebras (Huebschmann, 1990). A Lie algebroid on a manifold M is a triple A = (A; [[ ; ]];#A) where A is a vector bundle on M , (SecA; [[ ; ]]) is an R-Lie algebra, #A : A! TM is a linear homomorphism (called the anchor) of vector bundles and the following Leibniz condition is satis ed [[ ; f ]] = f [[ ; ]] + #A ( ) (f) ; f 2 C1 (M); ; 2 SecA: The anchor is bracket-preserving [B-K-W], [H] #A [[ ; ]] = [#A ;#A ]: A Lie algebroid is called transitive if #A is an epimorphism. For a transitive Lie algebroid A we have the Atiyah sequence 0 ! g ,!A #A ! TM ! 0;