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We extend results of Looijenga-Lunts and Verbitsky and show that the totalLie algebra $\mathfrak g$ for the intersection cohomology of a primitivesymplectic variety $X$ with isolated singularities is isomorphic to $$\mathfrakg \cong \mathfrak{so}((IH^2(X, \mathbb Q), Q_X)\oplus \mathfrak h)$$ where$Q_X$ is the intersection Beauville-Bogomolov-Fujiki form and $\mathfrak h$ isa hyperbolic plane. Along the way, we study the structure of $IH^*(X, \mathbbQ)$ as a $\mathfrak{g}$-representation -- with particular emphasis on theVerbitsky component, multidimensional Kuga-Satake constructions, andMumford-Tate algebras -- and give some immediate applications concerning the $P= W$ conjecture for primitive symplectic varieties.

The LLV Algebra for Primitive Symplectic Varieties with Isolated Singularities