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 $(\tilde{Q},\tilde{W})$ for which the CoHA is conjecturally the positive half of the Yangian $Y_{\text{MO}}({\mathfrak{g}}_Q)$. The goal of this article 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 $\text{KHA}$ and the $\text{CoHA}$ using the Chern character. As before, we expect our construction for the special class of quivers $(\tilde{Q},\tilde{W})$ to recover the positive part of quantum affine algebra $U_q(\widehat{{\mathfrak{g}}_Q})$ defined by Okounkov-Smirnov, but for general pairs $(Q,W)$ we expect new phenomena.

arXiv comment118 pages

References in corpus (13)

• Calabi-Yau algebras
• Base change for semiorthogonal decompositions
• A Lie algebra attached to a projective variety
• Characteristic classes in the Chow ring
• The cohomological Hall algebra of a preprojective algebra
• Combinatorial constructions of derived equivalences
• Two-dimensional categorified Hall algebras
• Moduli stacks of semistable sheaves and representations of Ext-quivers
• The integrality conjecture and the cohomology of preprojective stacks
• On two cohomological Hall algebras
• Donaldson-Thomas theory for categories of homological dimension one with potential
• On the Cohomological Hall Algebra of Dynkin quivers
• Deformed dimensional reduction

Cited by in corpus (7)

• Deformed dimensional reduction
• Derived categories of Quot schemes of locally free quotients via categorified Hall products
• Noncommutative resolutions and intersection cohomology for quotient singularities
• Generators for categorical Hall algebras of surfaces
• Intersection K-theory
• Shuffle algebras for quivers and R-matrices
• Hall-type algebras for categorical Donaldson-Thomas theories on local surfaces