We extend results of Looijenga-Lunts and Verbitsky and show that the total
Lie algebra g for the intersection cohomology of a primitive
symplectic variety X with isolated singularities is isomorphic to gβ so((IH2(X,Q),QXβ)βh) where
QXβ is the intersection Beauville-Bogomolov-Fujiki form and h is
a hyperbolic plane. Along the way, we study the structure of IHβ(X,Q) as a g-representation -- with particular emphasis on the
Verbitsky component, multidimensional Kuga-Satake constructions, and
Mumford-Tate algebras -- and give some immediate applications concerning the P=W conjecture for primitive symplectic varieties.Comment: 37 pages; Theorem 3.1 slightly adjusted; minor revisions for clarit