Symmetric self-adjoint Hopf categories and a categorical Heisenberg double

TL;DR

This paper introduces symmetric self-adjoint Hopf categories, constructs a categorical Heisenberg double, and provides categorifications of Heisenberg algebra representations using polynomial functors.

Contribution

It defines symmetric self-adjoint Hopf structures for semisimple categories and constructs a categorical Heisenberg double with applications to polynomial functors.

Findings

Categorification of Fock space representation of infinite Heisenberg algebra

Construction of categorical Heisenberg double with morphisms

Examples on polynomial and equivariant polynomial functors

Abstract

Motivated by the work of of A. Zelevinsky on positive self-adjoint Hopf algebras, we define what we call a symmetric self-adjoint Hopf structure for a certain kind of semisimple abelian categories. It is known that every positive self-adjoint Hopf algebra admits a natural action of the associated Heisenberg double. We construct canonical morphisms lifting the relations that define this action on the algebra level and define an object that we call a categorical Heisenberg double that is a natural setting for considering these morphisms. As examples, we exhibit the symmetric self-adjoint Hopf structure on the categories of polynomial functors and equivariant polynomial functors. In the case of the category of polynomial functors we obtain categorification of the Fock space representation of the infinite-dimensional Heisenberg algebra.