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.