K-theoretic Hall algebras for quivers with potential

Abstract

Given a quiver with potential (Q, W), Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of (Q, W). As shown by Davison-Meinhardt, this algebra comes with a filtration whose associated graded algebra is supercommutative. A special case of this construction is related to work of Nakajima, Varagnolo, Maulik-Okounkov etc. about geometric constructions of Yangians and their representations; indeed, given a quiver Q, there exists an associated pair (Q̃, W̃) for which the CoHA is conjecturally the positive half of the Yangian Y[subscript MO]([subscript gQ]). The goal of this thesis is to extend these ideas to K-theory. More precisely, we construct a K-theoretic Hall algebra using category of singularities, define a filtration whose associated graded algebra is a deformation of a symmetric algebra, and compare the KHA and the CoHA using the Chern character. As before, we expect our construction for the special class of quivers (Q̃, W̃) to recover the positive part of quantum affine algebra U[subscript q]([subscript gQ]) defined by Okounkov-Smirnov, but for general pairs (Q, W) we expect new phenomena.