Derived equivalences of hyperkähler varieties

We show that the Looijenga--Lunts--Verbitsky Lie algebra acting on the cohomology of a hyperkähler variety is a derived invariant, and obtain from this a number of consequences for the action on cohomology of derived equivalences between hyperkähler varieties. This includes a proof that derived equivalent hyperkähler varieties have isomorphic $\mathbf{Q}$-Hodge structures, the construction of a rational `Mukai lattice' functorial for derived equivalences, and the computation (up to index 2) of the image of the group of auto-equivalences on the cohomology of certain Hilbert squares of K3 surfaces.