Root datum: Difference between revisions

Revision as of 04:25, 25 September 2006

Revision as of 11:46, 25 September 2006

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:(''X''^*, Δ,''X''_*, Δ^v),

where

*''X''^* and ''X''_* are free abelian groupgroups 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

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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