# Root datum

In mathematical group theory, the **root datum** of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970.

## Definition

A **root datum** consists of a quadruple

- ,

where

- and are free abelian groups of finite rank together with a perfect pairing between them with values in which we denote by ( , ) (in other words, each is identified with the dual of the other).
- is a finite subset of and is a finite subset of and there is a bijection from onto , denoted by .
- For each , .
- For each , the map induces an automorphism of the root datum (in other words it maps to and the induced action on maps to )

The elements of
are called the **roots** of the root datum, and the elements of
are called the **coroots**.

If
does not contain
for any
, then the root datum is called **reduced**.

## The root datum of an algebraic group

If *G* is a reductive algebraic group over an algebraically closed field *K* with a split maximal torus *T* then its **root datum** is a quadruple

- (
*X*^{*}, Φ,*X*_{*}, Φ^{v}),

where

*X*^{*}is the lattice of characters of the maximal torus,*X*_{*}is the dual lattice (given by the 1-parameter subgroups),- Φ is a set of roots,
- Φ
^{v}is the corresponding set of coroots.

A connected split reductive algebraic group over *K* is uniquely determined (up to isomorphism) by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group.

For any root datum (*X*^{*}, Φ,*X*_{*}, Φ^{v}), we can define a **dual root datum** (*X*_{*}, Φ^{v},*X*^{*}, Φ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

If *G* is a connected reductive algebraic group over the algebraically closed field *K*, then its Langlands dual group ^{L}*G* is the complex connected reductive group whose root datum is dual to that of *G*.

## References

- Michel Demazure, Exp. XXI in SGA 3 vol 3
- T. A. Springer,
*Reductive groups*, in*Automorphic forms, representations, and L-functions*vol 1 ISBN 0-8218-3347-2