The LLV Algebra for Primitive Symplectic Varieties with Isolated Singularities

We extend results of Looijenga–Lunts and Verbitsky and show that the total Lie algebra $\mathfrak{g}$ for the intersection cohomology of a primitive symplectic variety $X$ with isolated singularities is isomorphic to $$\mathfrak{g}\cong \mathfrak{s}\mathfrak{o}((I{H}^2(X,\mathbb{Q}),Q_X)\oplus \mathfrak{h}),$$ where $Q_X$ is the intersection Beauville–Bogomolov–Fujiki form and $\mathfrak{h}$ is a hyperbolic plane. This gives a new, algebraic proof for irreducible holomorphic symplectic manifolds which does not rely on the hyperkähler metric. Along the way, we study the structure of $I{H}^{\ast }(X,\mathbb{Q})$ as a $\mathfrak{g}$-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.