Categorification and Heisenberg doubles arising from towers of algebras

TL;DR

This paper explores the categorification of Heisenberg doubles derived from towers of algebras, unifying existing theories and introducing new algebraic structures like quasi-Heisenberg algebras, with applications to symmetric functions.

Contribution

It develops the representation theory of Heisenberg doubles from towers of algebras and introduces quasi-Heisenberg algebras, expanding the framework of categorification.

Findings

Categorification of Fock space representations of Heisenberg doubles.

Introduction of quasi-Heisenberg algebras from 0-Hecke towers.

New proof that quasisymmetric functions are free over symmetric functions.

Abstract

The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the classical Heisenberg algebra. The aim of this paper is to study Heisenberg doubles arising from towers of algebras in this manner. First, we develop the basic representation theory of such Heisenberg doubles and show that if induction and restriction satisfy Mackey-like isomorphisms then the Fock space representation of the Heisenberg double has a natural categorification. This unifies the existing categorifications of the polynomial representation of the Weyl algebra and the Fock space representation of the Heisenberg algebra. Second, we develop in detail the theory applied to the tower of 0-Hecke algebras, obtaining new Heisenberg-like algebras that we call quasi-Heisenberg algebras. As an application of a generalized Stone--von Neumann Theorem, we give a new proof of the fact that the ring of quasisymmetric functions is free over the ring of symmetric functions.