Classification of the Mumford--Tate Groups of Rational Polarizable Hodge Structures

TL;DR

This paper characterizes the structure of Mumford--Tate groups associated with rational polarizable Hodge structures, revealing their relation to the Serre group and their decomposition into simple algebraic groups.

Contribution

It provides a detailed classification of Mumford--Tate groups by analyzing the structure of the associated pro-algebraic group G and its quotients, including the Serre group.

Findings

The quotient of G by its derived group is the Serre group.

The derived group of G is the simply connected cover of its adjoint group.

The adjoint group G decomposes into a product of specific simple algebraic groups.

Abstract

Let G be the pro-algebraic group attached to the tannakian category of polarizable rational Hodge structures. We show that the quotient of G by its derived group is the Serre group, the derived group of G is the simply connected covering of the adjoint group of G, and that the adjoint group G is a product of specific simple algebraic groups. As the Mumford--Tate groups are exactly the algebraic quotients of G, this also describes them.