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.Comment: 30 pages. v2: Minor changes. References adde

Representation theory of 0-Hecke–Clifford algebras

“…That guarantees both of the Grothendieck groups K and G to be Hopf algebras via the induction and the restriction. One can also refer to [11,Theorem 2.7], [24,Prop. 4.3] for the case of Hecke algebras.…”

“…Application: Peak * is free over Ω. Finally we consider the Heisenberg double arising from ( K, G) in order to prove that Peak * is a free Ω-module using the method of Savage et al in [24]. In general, given a graded Hopf pair (K(A), G(A)), there exists a left action of K(A) on G(A) such that G(A) is a K(A)-module algebra.…”

Towers of graded superalgebras categorify the twisted Heisenberg double

“…The goal of the current paper is to extend the results of [SY15] to the setting of towers of graded superalgebras. Since graded algebras and superalgebras are both ubiquitous in the categorification literature, the extension to this setting is a natural one.…”

Toward a polynomial basis of the algebra of peak quasisymmetric functions

“…In [19], Savage and Yacobi proved the freeness of QSym over its subring Λ of symmetric functions, alternatively using the technique of representation theory, namely, Heisenberg doubles arising from the tower of 0-Hecke algebras. Later, in [16] we applied such method to the case of PQSym by the tower of 0-Hecke-Clifford algebras, in order to prove the freeness of PQSym over its subring Γ spanned by Schur's Q-functions.…”