  Let X be a smooth complex projective variety. The group of autoequivalences
of the derived category of X acts naturally on its singular cohomology H(X, Q)
and we denote by Geq(X) the Zariski closure of its image in Gl(H(X, Q)). We
study the relation of the Lie algebra LieGeq(X) and the Neron-Severi Lie
algebra gNS(X) in case X has trivial canonical line bundle. At the same time
for mirror symmetric families of (weakly) Calabi-Yau varieties we consider a
conjecture of Kontsevich on the relation between the monodromy of one family
and the group Geq(X) for a very general member X of the other family.
