The cohomological Hall algebra of a preprojective algebra

Abstract

We introduce for each quiver Q and each algebraic oriented cohomology theory A, the cohomological Hall algebra (CoHA) of Q, as the A-homology of the moduli of representations of the preprojective algebra of Q. This generalizes the K-theoretic Hall algebra of commuting varieties defined by Schiffmann–Vasserot [‘The elliptic Hall algebra and the K-theory of the Hilbert scheme of A2, ’, Duke Math. J. 162 (2013) 279–366.]. When A is the Morava K-theory, we show evidence that this algebra is a candidate for Lusztig's reformulated conjecture on modular representations of algebraic groups [Lusztig, ‘On the character of certain irreducible modular representations’, Represent. Theory 19 (2015) 3–8; Joint Seminar, Mathematical Sciences Research Institute, October 28, 2014]. We construct an action of the preprojective CoHA on the A-homology of Nakajima quiver varieties. We compare this with the action of the Borel subalgebra of Yangian when A is the intersection theory. We also give a shuffle algebra description of this CoHA in terms of the underlying formal group law of A. As applications, we obtain a shuffle description of the Yangian.