In mathematics, the root datum (donnée radicielle in French) of a connected split reductive

Reductive group

In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...

 algebraic group

Algebraic group

In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

 over a field is a generalization of a root system

Root system

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...

 that determines the group up to isomorphism. They were introduced by Michel Demazure

Michel Demazure

Michel Demazure is a French mathematician. He made contributions in the fields of abstract algebra and algebraic geometry, was president of the French Mathematical Society and directed two French science museums.-Biography:...

 in SGA III

Grothendieck's Séminaire de géométrie algébrique

In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris...

, published in 1970.

Definition

A root datum consists of a quadruple,
where

• X^* and X_* are free abelian groups of finite rank
  Rank (linear algebra)
  The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
  
   together with a perfect pairing between them with values in Z which we denote by ( ) (in other words, each is identified with the dual lattice of the other).
• Δ is a finite subset of X^* and Δ^v is a finite subset of X_* and there is a bijection from Δ onto Δ^v, denoted by α→α^v.
• For each α, (α, α^v)=2
• For each α, the map taking x to x−(x,α^v)α induces an automorphism of the root datum (in other words it maps Δ to Δ and the induced action on X_* maps Δ ^v to Δ^v)

The elements of Δ are called the roots of the root datum, and the elements of Δ^v are called the coroots.

If Δ does not contain 2α for any α in Δ then the root datum is called reduced.

The root datum of an algebraic group

If G is a reductive algebraic group over a field K with a split maximal torus T then its root datum is a quadruple,
where

• X^* is the lattice of characters of the maximal torus,
• X_* is the dual lattice (given by the 1-parameter subgroups),
• Δ is a set of roots,
• Δ^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^*, Δ,X_*, Δ^v), we can define a dual root datum (X_*, Δ^v,X^*, Δ) 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 ^LG is the complex connected reductive group whose root datum is dual to that of G.