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Root datum

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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.

Contents

Definition

A root datum consists of a quadruple

( X , Φ , X , Φ ) ,

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 is a finite subset of X and there is a bijection from Φ onto Φ , denoted by α α .
  • For each α , ( α , α ) = 2 .
  • For each α , the map x x ( x , α ) α induces an automorphism of the root datum (in other words it maps Φ to Φ and the induced action on X maps Φ to Φ )
  • The elements of Φ are called the roots of the root datum, and the elements of Φ are called the coroots.

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

    The root datum of an algebraic group

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

    (X*, Φ, X*, Φv),

    where

  • X* is the lattice of characters of the maximal torus,
  • X* is the dual lattice (given by the 1-parameter subgroups),
  • Φ is a set of roots,
  • Φv 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*, Φ,X*, Φv), we can define a dual root datum (X*, Φv,X*, Φ) 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.

    References

    Root datum Wikipedia


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