A general approach to Heisenberg categorification via wreath product algebras

Abstract

We associate a monoidal category $${\mathcal {H}}_B$$ , 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 $${\mathfrak {h}}_B$$ . We show that, provided B is not concentrated in degree zero, the Grothendieck group of $${\mathcal {H}}_B$$ is isomorphic, as an algebra, to $${\mathfrak {h}}_B$$ . 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 $${\mathcal {H}}_B$$ contain generalizations of the degenerate affine Hecke algebra. Specializing B, this proves an open conjecture of Cautis–Licata.