We construct two Hirzebruch signature operators for some transitive invariantly oriented Lie algebroids A. Introducting a Riemannian structure in A we can define the *-Hodge operator, the codifferential and the first signature operator. Using the Hochschild-Serre spectral sequence for A and the Lusztig and Gromov examples we can define a second signature operator.
For each transitive Lie algebroid A over m-dimensional compact oriented manifold M and n-dimensional structure Lie algebras [then rankA=m+n ] such that the top cohomology group of A is not trivial, and if m+n=4p then the signature of A is defined. The problem is:
• to calculate the signature of A and give some conditions to the equality and vanishing of the signature.
Firstly, we give a general mechanism of the calculation of the signature via spectral sequences (Kubarski-Mishchenko 2003). Namely, under some simple regularity assumptions on a decreasing filtration on a DG-algebra C we have: if the second term leave in a finite rectangular and is a Poincaré algebra then the signature of the second term is equal to the signature of C.
We use this mechanism to
(a) the spectral sequence for the Čech-de Rham complex of the Lie algebroid A proving (for example) that if the structure Lie algebra g is a simple algebra of the type B_{k},C_{k},E7,E8,F4,G2 then the signature is zero.
(b) the Hochschild-Serre spectral sequence proving that the second term yields the Lusztig and Gromov examples.
Secondly, we describe a general algebraic approach to the *-Hodge operator, Hodge Theorem and Hirzebruch operator. We give applications to four fundamental examples concerning signature: classical example, Lie algebroids example, Lusztig example and Gromov example.