Root datum: Difference between revisions

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Revision as of 04:25, 25 September 2006

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==Definition==

A '''root datum''' consists of a quadruple

:(''X''^*, Δ,''X''_*, Δ^v),

where

*''X''^* and ''X''_* are free abelian group of finite rank together with a perfect pairing between them with values in '''Z''' (in other words, each is identified with the dual lattice of the other).

*Δ is a finite subset of ''X''^* and Δ^v is a finite subset of ''X''_* and there is a bijection from Δ onto Δ^v, denoted by α→α^v.

*For each α, (α, α^v)=2

*For each α, the map taking ''x'' to ''x''−(''x'',α^v)α induces an automorphism of the root datum (in other words it maps Δ to Δ and the induced action on ''X''_* maps Δ ^v to Δ^v)

The elements of Δ are called the '''roots''' of the root datum, and the elements of Δ^v are called the '''coroots'''.

If Δ does not contain 2α for any α in Δ then the root datum is called '''reduced'''.

==The root datum of an algebraic group==

If ''G'' is a reductive algebraic group over a field ''K'' with a split maximal torus ''T'' then its '''root datum''' is a quadruple

:(''X''^*, Δ,''X''_*, Δ^v),

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*Δ^v 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''^*, Δ,''X''_*, Δ^v), we can define a '''dual root datum''' (''X''_*, Δ^v,''X''^*, Δ) by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.

==References==