Root datum: Difference between revisions

Revision as of 02:02, 24 February 2007

Revision as of 02:03, 24 February 2007

Line 2:

==Definition==

A '''root datum''' consists of a quadruple (''X'', Ψ, ''X''^∨, Ψ^∨), where:

*<math>''X'', ''X^{\vee}''<sup>&or;</mathsup> are free abelian groups of finite [[rank]] together with a perfect pairing <math>\langle , \rangle : X \times X^{\vee} \rightarrow \mathbf{Z}</math> between them (in other words, each is identified with the [[dual lattice]] of the other).

* <math>\Psi </math> is a finite subset of <math>X</math> and <math>\Psi^{\vee} </math> is a finite subset of <math>X^{\vee}</math> and there is a bijection from <math>\Psi</math> onto <math>\Psi^{\vee}</math>, denoted by <math>\alpha \mapsto \alpha^{\vee}</math>.