Root datum: Difference between revisions

Revision as of 12:11, 13 January 2007

Revision as of 12:15, 13 January 2007

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==Definition==

A '''root datum''' consists of a quadruple <math>(X, \Psi,X^{\vee}, \Psi^{\vee}) </math>, where

*<math>X</math>, and <math>X^{\vee} </math> are free abelian groups of finite [[rank]] together with a perfect pairing between<math>\langle them, with\rangle values: inX <math>\times X^{\vee} \rightarrow \mathbf{Z} </math> between them (in other words, each is identified with the [[dual lattice]] of the other).

* <math>\Psi </math> is a finite subset of <math>X</math> and <math>\Psi^{\vee} </math> is a finite subset of <math>X^{\vee}</math> and there is a bijection from <math>\Psi</math> onto <math>\Psi^{\vee}</math>, denoted by &alpha;&rarr;&alpha;^v.

*For each &alpha;, (&alpha;, &alpha;^v)=2

* <math>\Psi </math> is a finite subset of <math>X</math> and <math>\Psi^{\vee} </math> is a finite subset of <math>X^{\vee}</math> and there is a bijection from <math>\Psi</math> onto <math>\Psi^{\vee}</math>, denoted by &alpha;&rarr;&alpha;<supmath>v\alpha \mapsto \alpha^{\vee}</supmath>.

*For each <math>\alpha</math>, we have: <math> \langle \alpha, \alpha^{\vee}\rangle =2 </math>

*For each &alpha;, the map taking ''x'' to ''x''&minus;(''x'',&alpha;^v)&alpha; induces an automorphism of the root datum (in other words it maps &Delta; to &Delta; and the induced action on ''X''_* maps &Delta; ^v to &Delta;^v)

The elements of &Delta;<math>\Psi</math> are called the '''roots''' of the root datum, and the elements of &Delta;<supmath>v \Psi^{\vee} </supmath> are called the '''coroots'''.

If &Delta; does not contain 2&alpha; for any &alpha; in &Delta; then the root datum is called '''reduced'''.