For a given symmetric quiver Q, we define a supercommutative quadratic algebra AQ whose Poincaré 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 AQ
Nearby in the stack
its Koszul dual Lie superalgebra
gQ
. We prove that the motivic Donaldson-Thomas invariants of the quiver
Q
may be computed using the Poincaré series of a certain Lie subalgebra of
gQ
that can be described, using an action of the first Weyl algebra on
gQ
, as the kernel of the operator
∂t
. This gives a new proof of positivity for motivic Donaldson--Thomas invariants. In addition, we prove that the algebra
AQ
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.