Hopf categories and the categorification of the Heisenberg algebra via graphical calculus

TL;DR

This paper investigates how Hopf categories can be used to categorify the infinite-dimensional Heisenberg algebra through graphical calculus, establishing a link between algebraic structures and categorical actions.

Contribution

It demonstrates that a Hopf structure on a semisimple symmetric monoidal abelian category induces a categorical action, advancing the categorification of the Heisenberg algebra.

Findings

Hopf structure implies categorical action

Strong categorification of the Heisenberg algebra achieved

Connection between Hopf categories and graphical calculus established

Abstract

We explore the connection between the notion of Hopf category and the categorification of the infinite dimensional Heisenberg algebra via graphical calculus proposed by M.Khovanov. We show that the existence of a Hopf structure on a semisimple symmetric monoidal abelian category implies existence of a categorical action in the sense of Khovanov and thus leads to a strong categorification of this algebra.