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Koszul algebras and Donaldson–Thomas invariants
We propose a new method for computing motivic Donaldson–Thomas invariants of a symmetric quiver which relies on Koszul duality between supercommutative algebras and Lie superalgebras and completely bypasses cohomological Hall algebras. Specifically, we define, for a given symmetric quiver Q, a supercommutative quadratic algebra A_Q, and study the Lie superalgebra g_Q that corresponds to A_Q under Koszul duality. We introduce an action of the first Weyl algebra on g_Q and prove that the motivic Donaldson–Thomas invariants of Q may be computed via the Poincaré series of the kernel of the operator ∂t. This gives a new proof of positivity for motivic Donaldson–Thomas invariants. Along the way, we prove that the algebra A_Q is numerically Koszul for every symmetric quiver Q and conjecture that it is in fact Koszul; we show that this conjecture holds for a certain class of quivers.