Towers of graded superalgebras categorify the twisted Heisenberg double

TL;DR

This paper demonstrates how towers of graded superalgebras can be used to categorify twisted Heisenberg doubles and quantum Weyl algebras through Grothendieck groups and functorial constructions.

Contribution

It introduces a framework for categorifying twisted Heisenberg doubles using towers of graded superalgebras and verifies this for specific algebra towers like wreath product and Sergeev superalgebras.

Findings

Grothendieck groups form twisted dual Hopf algebras

Categorification of twisted Heisenberg double and Fock space

Wreath product and Sergeev superalgebra towers satisfy the axioms

Abstract

We show that the Grothendieck groups of the categories of finitely-generated graded supermodules and finitely-generated projective graded supermodules over a tower of graded superalgebras satisfying certain natural conditions give rise to twisted Hopf algebras that are twisted dual. Then, using induction and restriction functors coming from such towers, we obtain a categorification of the twisted Heisenberg double and its Fock space representation. We show that towers of wreath product algebras (in particular, the tower of Sergeev superalgebras) and the tower of nilcoxeter graded superalgebras satisfy our axioms. In the latter case, one obtains a categorification of the quantum Weyl algebra.