The Hodge Lie algebra acting on cohomology of a smooth projective variety was introduced 
in a joint paper with Eduard Looijenga 20 years ago ("A Lie algebra attached to a projective variety"). In a subsequent paper with Golyshev and Orlov "Mirror symmetry for abelian varieties" it was shown that for an abelian variety A this Lie algebra is closely related to the group of autoequivalences of the derived category D^b(cohA). I will report on that and then suggest a possible extension of the results to the case of hyperkahler manifolds.