Compact Lie Groups and Complex Reductive Groups

TL;DR

This paper establishes a homotopy equivalence between the categories of compact Lie groups and complex reductive groups, revealing a deep topological connection and equivalence at the level of infinity categories.

Contribution

It demonstrates that the categories of compact Lie groups and complex reductive groups are homotopy equivalent, providing a new perspective on their topological and categorical relationship.

Findings

Categories are homotopy equivalent topological categories

Equivalence extends to infinity categories

Deepens understanding of the relationship between Lie and reductive groups

Abstract

We show that the categories of compact Lie groups and complex reductive groups (not necessarily connected) are homotopy equivalent topological categories. In other words, the corresponding categories enriched in the homotopy category of topological spaces are equivalent. This can also be interpreted as an equivalence of infinity categories.