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.

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**.

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*_{*}, Φ

^{v}),

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,
Φ^{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*^{*}, Φ,*X*_{*}, Φ^{v}), we can define a **dual root datum** (*X*_{*}, Φ^{v},*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 ^{L}*G* is the complex connected reductive group whose root datum is dual to that of *G*.