Reference request: Conjugacy classes of real semi-simple Lie group $G$ and their decomposition into...
$begingroup$
Let $G$ be a real semi-simple Lie group and $K$ its maximal compact subgroup.
Is there a description of the orbit space of the conjugation action on $G$ by itself?
Is there a decription of how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$?
I would appreciate it if anyone could point out any references.
reference-request lie-groups
asked Jan 9 at 11:57
No_wayNo_way
59118
$endgroup$
add a comment |
$begingroup$
Let $G$ be a real semi-simple Lie group and $K$ its maximal compact subgroup.
Is there a description of the orbit space of the conjugation action on $G$ by itself?
Is there a decription of how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$?
I would appreciate it if anyone could point out any references.
reference-request lie-groups
asked Jan 9 at 11:57
No_wayNo_way
59118
$endgroup$
$begingroup$
I do not understand the second question but the first is definitely no, just think about the Jordan normal form. JNF does generalize to other semisimple Lie groups.
$endgroup$
– Moishe Kohan
Jan 11 at 1:40
$begingroup$
Thanks for your comments on item 1, which has just been edited. For item 2, I was asking for how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$.
$endgroup$
– No_way
Jan 12 at 12:43
add a comment |
$begingroup$
Let $G$ be a real semi-simple Lie group and $K$ its maximal compact subgroup.
Is there a description of the orbit space of the conjugation action on $G$ by itself?
Is there a decription of how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$?
I would appreciate it if anyone could point out any references.
reference-request lie-groups
asked Jan 9 at 11:57
No_wayNo_way
59118
$endgroup$
Let $G$ be a real semi-simple Lie group and $K$ its maximal compact subgroup.
Is there a description of the orbit space of the conjugation action on $G$ by itself?
Is there a decription of how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$?
I would appreciate it if anyone could point out any references.
reference-request lie-groups
reference-request lie-groups
asked Jan 9 at 11:57
No_wayNo_way
59118
asked Jan 9 at 11:57
No_wayNo_way
59118
edited Jan 12 at 12:37
No_way
asked Jan 9 at 11:57
No_wayNo_way
59118
asked Jan 9 at 11:57
No_wayNo_way
59118
asked Jan 9 at 11:57
No_wayNo_way
59118
59118
$begingroup$
I do not understand the second question but the first is definitely no, just think about the Jordan normal form. JNF does generalize to other semisimple Lie groups.
$endgroup$
– Moishe Kohan
Jan 11 at 1:40
$begingroup$
Thanks for your comments on item 1, which has just been edited. For item 2, I was asking for how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$.
$endgroup$
– No_way
Jan 12 at 12:43
add a comment |
$begingroup$
I do not understand the second question but the first is definitely no, just think about the Jordan normal form. JNF does generalize to other semisimple Lie groups.
$endgroup$
– Moishe Kohan
Jan 11 at 1:40
$begingroup$
Thanks for your comments on item 1, which has just been edited. For item 2, I was asking for how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$.
$endgroup$
– No_way
Jan 12 at 12:43
$begingroup$
I do not understand the second question but the first is definitely no, just think about the Jordan normal form. JNF does generalize to other semisimple Lie groups.
$endgroup$
– Moishe Kohan
Jan 11 at 1:40
$begingroup$
I do not understand the second question but the first is definitely no, just think about the Jordan normal form. JNF does generalize to other semisimple Lie groups.
$endgroup$
– Moishe Kohan
Jan 11 at 1:40
$begingroup$
Thanks for your comments on item 1, which has just been edited. For item 2, I was asking for how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$.
$endgroup$
– No_way
Jan 12 at 12:43
$begingroup$
Thanks for your comments on item 1, which has just been edited. For item 2, I was asking for how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$.
$endgroup$
– No_way
Jan 12 at 12:43
add a comment |
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$begingroup$
I do not understand the second question but the first is definitely no, just think about the Jordan normal form. JNF does generalize to other semisimple Lie groups.
$endgroup$
– Moishe Kohan
Jan 11 at 1:40
$begingroup$
Thanks for your comments on item 1, which has just been edited. For item 2, I was asking for how a conjugacy class of $G$ decomposes into orbits of the conjugation action by $K$.
$endgroup$
– No_way
Jan 12 at 12:43