Quantum Frobenius Heisenberg categorification

TL;DR

This paper introduces a new diagrammatic monoidal category called the quantum Frobenius Heisenberg category, which generalizes existing categories and connects to skein categories, with a basis theorem for morphism spaces.

Contribution

It constructs the quantum Frobenius Heisenberg category associated to a symmetric Frobenius superalgebra, extending previous quantum Heisenberg categories and affine HOMFLY-PT skein categories.

Findings

Established a basis theorem for morphism spaces.

Connected the new category to existing quantum and skein categories.

Provided categorical actions on generalized cyclotomic quotients.

Abstract

We associate a diagrammatic monoidal category $\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius Heisenberg category, to a symmetric Frobenius superalgebra $A$, a central charge $k \in \mathbb{Z}$, and invertible parameters $z,t$ in some ground ring. When $A$ is trivial, i.e. it equals the ground ring, these categories recover the quantum Heisenberg categories introduced in our previous work, and when the central charge $k$ is zero they yield generalizations of the affine HOMFLY-PT skein category. By exploiting some natural categorical actions of $\mathcal{H}\textit{eis}_k(A;z,t)$ on generalized cyclotomic quotients, we prove a basis theorem for morphism spaces.