Classification of reductive groups via root datum

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• The Tortoise-Man
The Tortoise-Man
I have a couple of questions about classification of reductive groups over algebraically closed field (up to isomorphism) by so called root datum.

In the linked discussion is continued that

In particular, the semisimple groups over an algebraically closed field
are classified up to central isogenies by their Dynkin diagrams.

Obviously, a root datum ##(X^*, \Phi, X_*, \phi^{\vee})## contains full information ("building plan") about the associated Dynkin diagram, but the converse is not true: A root datum contains slightly more information than the Dynkin diagram, eg it "knows" the center of the given reductive group.

Questions:

1) How concretely a root datum ##(X^*(T), \Phi, X(T)_*, \phi^{\vee})## associated to a reductive group ##G## with maximal torus ##T## "reconstructs" fully the center ##Z(G)## of the group? (in other words: why does this root datum "contain" full information about the
center of this group?

2) The quoted statement above claims that due to this classification of reductive groups via root data, the semisimple groups - which form a subclass of reductive groups, those with ##R(G)=1## - are classified up to central isogenies by their Dynkin diagrams.

But aren't then in turn all reductive groups, not only the semisimple ones classified - up to central isogenies! - by their Dynkin diagrams?

Because, isn't the quotient map ##G \to G/Z(G)## always an isogeny, or
is this quotient map only an isogeny when ##G## is semisimple?

The question in 2) at all becomes finally "how much more information" do a root datum contain than the associated Dynkin diagram only? Problem in 1) suggests that the only "additional piece" of information which the root datum carries but the
Dynkin diagram "not sees", is the information about the center of the group.

Last edited:
The Tortoise-Man said:
I have a couple of questions about classification of reductive groups over algebraically closed field (up to isomorphism) by so called root datum.

In the linked discussion is continued that
Obviously, a root datum ##(X^*, \Phi, X_*, \phi^{\vee})## contains full information ("building plan") about the associated Dynkin diagram, but the converse is not true: A root datum contains slightly more information than the Dynkin diagram, eg it "knows" the center of the given reductive group.

Questions:

1) How concretely a root datum ##(X^*(T), \Phi, X(T)_*, \phi^{\vee})## associated to a reductive group ##G## with maximal torus ##T## "reconstructs" fully the center ##Z(G)## of the group? (in other words: why does this root datum "contain" full information about the
center of this group?
For each root, consider the connocted component of the kernel, then the intersecrion of all these kernels, then take the connected component. That is the center.
The Tortoise-Man said:
2) The quoted statement above claims that due to this classification of reductive groups via root data, the semisimple groups - which form a subclass of reductive groups, those with ##R(G)=1## - are classified up to central isogenies by their Dynkin diagrams.

But aren't then in turn all reductive groups, not only the semisimple ones classified - up to central isogenies! - by their Dynkin diagrams?

Because, isn't the quotient map ##G \to G/Z(G)## always an isogeny, or
is this quotient map only an isogeny when ##G## is semisimple?
No, because to be an isogeny the kernel has to be finite. Here the kernel is the center, which need not be finite.
The Tortoise-Man said:
The question in 2) at all becomes finally "how much more information" do a root datum contain than the associated Dynkin diagram only? Problem in 1) suggests that the only "additional piece" of information which the root datum carries but the
Dynkin diagram "not sees", is the information about the center of the group.

Alright, so only the semisimple ones are precisely those with finite center, that's the issue, right?

The Tortoise-Man said:
Alright, so only the semisimple ones are precisely those with finite center, that's the issue, right?
Yes. Think of ##GL_n## is reductive and ##SL_n## semisimple.

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