The cohomological Hall algebra of a preprojective algebra

TL;DR

This paper introduces a cohomological Hall algebra for quivers using algebraic oriented cohomology theories, generalizing previous work and connecting to modular representations, Nakajima quiver varieties, and Yangians.

Contribution

It constructs a new cohomological Hall algebra for preprojective algebras, extends existing theories, and explores its actions and descriptions in various algebraic contexts.

Findings

Defined the cohomological Hall algebra for quivers and preprojective algebras.

Established connections with Nakajima quiver varieties and Yangian actions.

Provided a shuffle algebra description related to formal group laws.

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. 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. 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.