- Source: Root datum
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
A root datum consists of a quadruple
(
X
∗
,
Φ
,
X
∗
,
Φ
∨
)
{\displaystyle (X^{\ast },\Phi ,X_{\ast },\Phi ^{\vee })}
,
where
X
∗
{\displaystyle X^{\ast }}
and
X
∗
{\displaystyle X_{\ast }}
are free abelian groups of finite rank together with a perfect pairing between them with values in
Z
{\displaystyle \mathbb {Z} }
which we denote by ( , ) (in other words, each is identified with the dual of the other).
Φ
{\displaystyle \Phi }
is a finite subset of
X
∗
{\displaystyle X^{\ast }}
and
Φ
∨
{\displaystyle \Phi ^{\vee }}
is a finite subset of
X
∗
{\displaystyle X_{\ast }}
and there is a bijection from
Φ
{\displaystyle \Phi }
onto
Φ
∨
{\displaystyle \Phi ^{\vee }}
, denoted by
α
↦
α
∨
{\displaystyle \alpha \mapsto \alpha ^{\vee }}
.
For each
α
{\displaystyle \alpha }
,
(
α
,
α
∨
)
=
2
{\displaystyle (\alpha ,\alpha ^{\vee })=2}
.
For each
α
{\displaystyle \alpha }
, the map
x
↦
x
−
(
x
,
α
∨
)
α
{\displaystyle x\mapsto x-(x,\alpha ^{\vee })\alpha }
induces an automorphism of the root datum (in other words it maps
Φ
{\displaystyle \Phi }
to
Φ
{\displaystyle \Phi }
and the induced action on
X
∗
{\displaystyle X_{\ast }}
maps
Φ
∨
{\displaystyle \Phi ^{\vee }}
to
Φ
∨
{\displaystyle \Phi ^{\vee }}
)
The elements of
Φ
{\displaystyle \Phi }
are called the roots of the root datum, and the elements of
Φ
∨
{\displaystyle \Phi ^{\vee }}
are called the coroots.
If
Φ
{\displaystyle \Phi }
does not contain
2
α
{\displaystyle 2\alpha }
for any
α
∈
Φ
{\displaystyle \alpha \in \Phi }
, then the root datum is called reduced.
The root datum of an algebraic group
If
G
{\displaystyle G}
is a reductive algebraic group over an algebraically closed field
K
{\displaystyle K}
with a split maximal torus
T
{\displaystyle T}
then its root datum is a quadruple
(
X
∗
,
Φ
,
X
∗
,
Φ
∨
)
{\displaystyle (X^{*},\Phi ,X_{*},\Phi ^{\vee })}
,
where
X
∗
{\displaystyle X^{*}}
is the lattice of characters of the maximal torus,
X
∗
{\displaystyle X_{*}}
is the dual lattice (given by the 1-parameter subgroups),
Φ
{\displaystyle \Phi }
is a set of roots,
Φ
∨
{\displaystyle \Phi ^{\vee }}
is the corresponding set of coroots.
A connected split reductive algebraic group over
K
{\displaystyle 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
∗
,
Φ
∨
)
{\displaystyle (X^{*},\Phi ,X_{*},\Phi ^{\vee })}
, we can define a dual root datum
(
X
∗
,
Φ
∨
,
X
∗
,
Φ
)
{\displaystyle (X_{*},\Phi ^{\vee },X^{*},\Phi )}
by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If
G
{\displaystyle G}
is a connected reductive algebraic group over the algebraically closed field
K
{\displaystyle K}
, then its Langlands dual group
L
G
{\displaystyle {}^{L}G}
is the complex connected reductive group whose root datum is dual to that of
G
{\displaystyle G}
.
References
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
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