The cohomological Hall algebras of a preprojective algebra with symmetrizer

TL;DR

This paper constructs a geometric realization of Yangians for non-simply laced types using cohomological Hall algebras of quivers with symmetrizers, establishing connections with generalized preprojective algebras and shuffle formulas.

Contribution

It introduces a new geometric framework for Yangians via cohomological Hall algebras of quivers with symmetrizers, including a shuffle formula and relation to generalized preprojective algebras.

Findings

Proves a dimensional reduction for the cohomological Hall algebra.

Provides a shuffle formula for the algebra.

Shows the algebra satisfies Yangian relations for symmetrizable Cartan matrices.

Abstract

This paper aims at a geometric realization of the Yangian of non-simply laced type in terms of quiver with potentials. For every quiver with symmetrizer, there is an extended quiver with superpotential, whose Jacobian algebra is the generalized preprojective algebra of Gei{\ss}, Leclerc, and Schr\"oer arXiv:1410.1403. We study the cohomological Hall algebra of Kontsevich and Soibelman associated to this quiver with potential. In particular, we prove a dimensional reduction result, and provide a shuffle formula of this cohomological Hall algebra. In the case when the quiver with symmetrizer comes from a symmetrizable Cartan matrix, we prove that this shuffle algebra satisfies the relations of the Yangian associated to this Cartan matrix.