Root datum

(Redirected from Root data)

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

edit

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

edit

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

 ,

where

  •   is the lattice of characters of the maximal torus,
  •   is the dual lattice (given by the 1-parameter subgroups),
  •   is a set of roots,
  •   is the corresponding set of coroots.

A connected split reductive algebraic group over   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  , we can define a dual root datum   by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

If   is a connected reductive algebraic group over the algebraically closed field  , then its Langlands dual group   is the complex connected reductive group whose root datum is dual to that of  .

References

edit
  • 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