Categorical Tori

TL;DR

This paper constructs explicit categorical extensions of tori by circles, explores their applications to loop group extensions, and provides examples involving Lie groups and lattices, revealing new insights into their structure and fixed points.

Contribution

It introduces elementary constructions of categorical extensions of tori, applies them to loop groups, and characterizes extraspecial 2-groups via categorical fixed points.

Findings

Explicit constructions of categorical tori and their extensions.

Applications to loop group extensions and lattice examples.

Identification of extraspecial 2-groups as categorical fixed points.

Abstract

We give explicit and elementary constructions of the categorical extensions of a torus by the circle and discuss an application to loop group extensions. Examples include maximal tori of simple and simply connected compact Lie groups and the tori associated to the Leech and Niemeyer lattices. We obtain the extraspecial 2-groups as the isomorphism classes of categorical fixed points under an involution action.