Quasi-connected reductive groups

TL;DR

This paper introduces quasi-connected reductive groups, characterizing them as almost direct products of semisimple groups and quasi-tori, and establishes their relation to smooth normal subgroups of connected reductive groups.

Contribution

It defines quasi-connected reductive groups over arbitrary fields and proves their equivalence to certain smooth normal subgroups of connected reductive groups.

Findings

Quasi-connected reductive groups are characterized as almost direct products.

Such groups are precisely the smooth normal subgroups of connected reductive groups.

The paper extends the understanding of reductive groups over arbitrary fields.

Abstract

We introduce the notion of a quasi-connected reductive group over an arbitrary field to be an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic group is quasi-connected reductive if and only if it is isomorphic to a smooth normal subgroup of a connected reductive group.