Mixing
Degree of universal cover of simple Lie group
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I have seen a statement that if $mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $mathfrak{g}$. Equivalently, the simply connected group with Lie algebra $mathfrak{g}$ has finite center. Equivalently, the fundamental group of a Lie group with Lie algebra $mathfrak{g}$ is finite.
Can anyone provide a proof or reference for any of these equivalent statements?
lie-groups lie-algebras covering-spaces fundamental-groups
asked Jan 16 at 18:00
user320832user320832
1707
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add a comment |
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I have seen a statement that if $mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $mathfrak{g}$. Equivalently, the simply connected group with Lie algebra $mathfrak{g}$ has finite center. Equivalently, the fundamental group of a Lie group with Lie algebra $mathfrak{g}$ is finite.
Can anyone provide a proof or reference for any of these equivalent statements?
lie-groups lie-algebras covering-spaces fundamental-groups
asked Jan 16 at 18:00
user320832user320832
1707
$endgroup$
$begingroup$
See this question and the linked questions and references, which together give the result.
$endgroup$
– Dietrich Burde
Jan 17 at 15:41
$begingroup$
Thanks for pointing me there! But the first answer is only for compact Lie groups, and I certainly hope I don’t need to use Ricci curvature like in the second answer. (And the links in the comments are broken.)
$endgroup$
– user320832
Jan 18 at 5:35
add a comment |
$begingroup$
I have seen a statement that if $mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $mathfrak{g}$. Equivalently, the simply connected group with Lie algebra $mathfrak{g}$ has finite center. Equivalently, the fundamental group of a Lie group with Lie algebra $mathfrak{g}$ is finite.
Can anyone provide a proof or reference for any of these equivalent statements?
lie-groups lie-algebras covering-spaces fundamental-groups
asked Jan 16 at 18:00
user320832user320832
1707
$endgroup$
I have seen a statement that if $mathfrak{g}$ is a simple Lie algebra, then there are only finitely many Lie groups with Lie algebra $mathfrak{g}$. Equivalently, the simply connected group with Lie algebra $mathfrak{g}$ has finite center. Equivalently, the fundamental group of a Lie group with Lie algebra $mathfrak{g}$ is finite.
Can anyone provide a proof or reference for any of these equivalent statements?
lie-groups lie-algebras covering-spaces fundamental-groups
lie-groups lie-algebras covering-spaces fundamental-groups
asked Jan 16 at 18:00
user320832user320832
1707
asked Jan 16 at 18:00
user320832user320832
1707
asked Jan 16 at 18:00
user320832user320832
1707
asked Jan 16 at 18:00
user320832user320832
1707
asked Jan 16 at 18:00
user320832user320832
1707
1707
$begingroup$
See this question and the linked questions and references, which together give the result.
$endgroup$
– Dietrich Burde
Jan 17 at 15:41
$begingroup$
Thanks for pointing me there! But the first answer is only for compact Lie groups, and I certainly hope I don’t need to use Ricci curvature like in the second answer. (And the links in the comments are broken.)
$endgroup$
– user320832
Jan 18 at 5:35
add a comment |
$begingroup$
See this question and the linked questions and references, which together give the result.
$endgroup$
– Dietrich Burde
Jan 17 at 15:41
$begingroup$
Thanks for pointing me there! But the first answer is only for compact Lie groups, and I certainly hope I don’t need to use Ricci curvature like in the second answer. (And the links in the comments are broken.)
$endgroup$
– user320832
Jan 18 at 5:35
$begingroup$
See this question and the linked questions and references, which together give the result.
$endgroup$
– Dietrich Burde
Jan 17 at 15:41
$begingroup$
See this question and the linked questions and references, which together give the result.
$endgroup$
– Dietrich Burde
Jan 17 at 15:41
$begingroup$
Thanks for pointing me there! But the first answer is only for compact Lie groups, and I certainly hope I don’t need to use Ricci curvature like in the second answer. (And the links in the comments are broken.)
$endgroup$
– user320832
Jan 18 at 5:35
$begingroup$
Thanks for pointing me there! But the first answer is only for compact Lie groups, and I certainly hope I don’t need to use Ricci curvature like in the second answer. (And the links in the comments are broken.)
$endgroup$
– user320832
Jan 18 at 5:35
add a comment |
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$begingroup$
See this question and the linked questions and references, which together give the result.
$endgroup$
– Dietrich Burde
Jan 17 at 15:41
$begingroup$
Thanks for pointing me there! But the first answer is only for compact Lie groups, and I certainly hope I don’t need to use Ricci curvature like in the second answer. (And the links in the comments are broken.)
$endgroup$
– user320832
Jan 18 at 5:35