A semisimple subcategory of Khovanov's Heisenberg category

TL;DR

This paper constructs a semisimple subcategory within Khovanov's Heisenberg category, revealing complex tensor-ideal structures and challenging existing classification methods in tensor-triangular geometry.

Contribution

It introduces a semisimple replete subcategory that preserves object isomorphisms and provides a novel example of a monoidal triangulated category with unique tensor-ideal properties.

Findings

Existence of a semisimple subcategory retaining object isomorphisms

Counterexample to the classification of one-sided thick tensor-ideals

Illustration of limitations of standard support varieties

Abstract

We show the existence of a semisimple replete subcategory of Khovanov's Heisenberg category that retains the isomorphism data of objects for the full category. This leads to a noncommutative tensor-triangular geometric example of a monoidal triangulated category whose Balmer spectrum satisfies the tensor product property but which contains one-sided thick tensor-ideals that are not two-sided, and whose standard support varieties fail to classify one-sided thick tensor-ideals.