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root datum : ウィキペディア英語版
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
:(X^\ast, \Phi, X_\ast, \Phi^\vee),
where
* X^\ast and X_\ast are free abelian groups of finite rank together with a perfect pairing between them with values in \mathbb which we denote by ( , ) (in other words, each is identified with the dual lattice of the other).
* \Phi is a finite subset of X^\ast and \Phi^\vee is a finite subset of X_\ast and there is a bijection from \Phi onto \Phi^\vee, denoted by \alpha\mapsto\alpha^\vee.
* For each \alpha, (\alpha, \alpha^\vee)=2.
* For each \alpha, the map x\mapsto x-(x,\alpha^\vee)\alpha induces an automorphism of the root datum (in other words it maps \Phi to \Phi and the induced action on X_\ast maps \Phi^\vee to \Phi^\vee)
The elements of \Phi are called the roots of the root datum, and the elements of \Phi^\vee are called the coroots. The elements of X^\ast are sometimes called weights and those of X_\ast accordingly coweights.
If \Phi does not contain 2\alpha for any \alpha\in\Phi, then the root datum is called reduced.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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