Root datum

Related on Lably

Definition

A root datum consists of a quadruple

(X^{*}, Φ, X_{*}, Φ^{\lor})

where

$X^{*}$ and $X_{*}$ are free abelian groups of finite rank together with a perfect pairing between them with values in $Z$ which we denote by ( , ) (in other words, each is identified with the dual of the other).
$Φ$ is a finite subset of $X^{*}$ and $Φ^{\lor}$ is a finite subset of $X_{*}$ and there is a bijection from $Φ$ onto $Φ^{\lor}$ , denoted by $α \mapsto α^{\lor}$ .
For each $α$ , $(α, α^{\lor}) = 2$ .
For each $α$ , the map $x \mapsto x - (x, α^{\lor}) α$ induces an automorphism of the root datum (in other words it maps $Φ$ to $Φ$ and the induced action on $X_{*}$ maps $Φ^{\lor}$ to $Φ^{\lor}$ )

The elements of $Φ$ are called the roots of the root datum, and the elements of $Φ^{\lor}$ are called the coroots.

If $Φ$ does not contain $2 α$ for any $α \in Φ$ , then the root datum is called reduced.

One more chapter is free with a Lably account.