Categorification and Heisenberg doubles arising from towers of algebras

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.

30 pages. v2: Minor changes. References added

References in corpus (3)

• Heisenberg categorification and Hilbert schemes
• Hecke algebras, finite general linear groups, and Heisenberg categorification
• A survey of Heisenberg categorification via graphical calculus

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• Introduction to categorification
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• Representation theory of 0-Hecke-Clifford algebras
• An equivalence between truncations of categorified quantum groups and Heisenberg categories
• Symmetric self-adjoint Hopf categories and a categorical Heisenberg double
• Indecomposable objects in Khovanov-Sazdanovic's generalizations of Deligne's interpolation categories