Disconnected reductive groups

TL;DR

This paper classifies disconnected complex reductive algebraic groups by establishing a bijection between such groups and algebraic extensions of their component group by the center of their identity component.

Contribution

It provides a new classification framework for disconnected reductive groups using algebraic extensions, linking group structure to algebraic data.

Findings

Classification of disconnected reductive groups via algebraic extensions

Establishment of a bijection between groups and extensions

Insight into the structure of component groups and centers

Abstract

In this paper, we describe the possible disconnected complex reductive algebraic groups $E$ with component group $\Gamma = E/E_0$. We show that there is a natural bijection between such groups $E$ and algebraic extensions of $\Gamma$ by $Z(E_0)$.