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.