K-theoretic Hall algebras

Abstract: Given a quiver with potential, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of $(Q,W)$. In particular cases, one recovers positive parts of Yangians as defined by Maulik-Okounkov. For general $(Q,W)$, the Hall algebra has nice structure properties, for example Davison-Meinhardt proved a PBW theorem for it using the decomposition theorem.

One can define a $K$-theoretic version of this algebra using certain categories of singularities that depend on the stack of representations of $(Q,W)$. In particular cases, these Hall algebras are positive parts of quantum affine algebras. We show that some of the structure properties in cohomology, such as the PBW theorem, can be lifted to $K$-theory, replacing the use of the decomposition theorem with semi-orthogonal decompositions.

mathematical physicsalgebraic geometryrepresentation theory

Audience: researchers in the topic

Geometric Representation Theory conference

Series comments: Originally planned as a twinned conference held simultaneously at the Max Planck Institute in Bonn, Germany and the Perimeter Institute in Waterloo, Canada. The concept was motivated by the desire to reduce the environmental impact of conference travels. In order to view the talks, register at the website: www.mpim-bonn.mpg.de/grt2020 . The talks from previous days can be be viewed at pirsa.org/C20030 ; slides from the talks are posted here: www.dropbox.com/sh/cjzqbqn7ql8zcjv/AAANB82Hh4t5XDc5RPcZzW0Aa?dl=0