Cohomological Hall algebra of a preprojective algebra and Yangian

This talk is about Hall algebras and quantum groups. By a theorem of
Ringel, the Hall algebra of a quiver is isomorphic to the positive
half of the quantum enveloping algebra of a Lie algebra. In this talk,

I will talk about one analog of the above theorem.

I will describe a cohomological Hall algebra of representation space
of preprojective algebra (preprojective CoHA for short) for any
quiver. This generalizes the elliptic Hall algebra of
Schiffmann-Vasserot. I will describe a family of representations of
the preprojective CoHA constructed out of homology of Nakajima quiver
varieties. We show the (spherical part of) preprojective CoHA is
isomorphic to the positive half of the Yangian, where the Yangian is a

deformation of the enveloping algebra of the current algebra of a Lie
algebra. I will also compare the preprojective CoHA action on homology

of Nakajima quiver varieties with the two Yangian actions constructed
by Varagnolo and Maulik-Okounkov.

If time permits, I will talk about the relation of the preprojective
CoHA and the critical CoHA for quiver with potentials defined by
Kontsevich-Soibelman in special cases.

This talk is based on my joint work with Gufang Zhao.