Commutativity of the nilpotent CoHAs of the affine plane

The cohomological Hall algebra associated with finite-length coherent sheaves on the affine plane has a rich structure, described in the unrestricted case by Davison. This lecture studies the variants obtained by restricting to sheaves whose set-theoretic support lies at the origin or along a coordinate axis. Both resulting nilpotent CoHAs are commutative. Equivariant deformation with respect to simultaneous scaling of the coordinates, however, gives rise to noncommutative algebras that can be described explicitly.