A general approach to Heisenberg categorification via wreath product algebras

Abstract

We associate a monoidal category HB, defined in terms of planar diagrams, to any graded Frobenius superalgebra B. This category acts naturally on modules over the wreath product algebras associated to B. To B we also associate a (quantum) lattice Heisenberg algebra hB. We show that, provided B is not concentrated in degree zero, the Grothendieck group of HB is isomorphic, as an algebra, to hB. For specific choices of Frobenius algebra B, we recover existing results, including those of Khovanov and Cautis–Licata. We also prove that certain morphism spaces in the category HB contain generalizations of the degenerate affine Hecke algebra. Specializing B, this proves an open conjecture of Cautis–Licata.