Koszul algebras and Donaldson-Thomas invariants

For a given symmetric quiver $Q$, we define a supercommutative quadratic algebra ${\mathcal{A}}_Q$ whose Poincar'e series is related to the motivic generating function of $Q$ by a simple change of variables. The Koszul duality between supercommutative algebras and Lie superalgebras assigns to the algebra ${\mathcal{A}}_Q$ its Koszul dual Lie superalgebra ${\mathfrak{g}}_Q$. We prove that the motivic Donaldson-Thomas invariants of the quiver $Q$ may be computed using the Poincar'e series of a certain Lie subalgebra of ${\mathfrak{g}}_Q$ that can be described, using an action of the first Weyl algebra on ${\mathfrak{g}}_Q$, as the kernel of the operator ${\mathrm{\partial }}_t$. This gives a new proof of positivity for motivic Donaldson–Thomas invariants. In addition, we prove that the algebra ${\mathcal{A}}_Q$ is numerically Koszul for every symmetric quiver $Q$ and conjecture that it is in fact Koszul; we also prove this conjecture for quivers of a certain class.