\(Q\)-algebroids and their cohomology. (English) Zbl 1215.22002

Mehta, Rajan Amit

J. Symplectic Geom. 7, No. 3, 263-293 (2009).

Summary: A \(Q\)-algebroid is a graded Lie algebroid equipped with a compatible homological vector field and is the infinitesimal object corresponding to a \(Q\)-groupoid. We associate to every \(Q\)-algebroid a double complex. As a special case, we define the Becchi-Rouet-Stora-Tyutin (BRST) model of a Lie algebroid, which generalizes the BRST model for equivariant cohomology. We extend to this setting the Mathai-Quillen-Kalkman isomorphism of the BRST and Weil models, and we suggest a definition of a basic subcomplex which, however, requires a choice of a connection. Other examples include Roytenberg’s homological double of a Lie bialgebroid, Ginzburg’s model of equivariant Lie algebroid cohomology, the double of a Lie algebroid matched pair, and \(Q\)-algebroids arising from lifted actions on Courant algebroids.

MSC:

22A22	Topological groupoids (including differentiable and Lie groupoids)
53D17	Poisson manifolds; Poisson groupoids and algebroids
55N91	Equivariant homology and cohomology in algebraic topology