Talk:Lie groups I: Definition and examples
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I'm going to add the condition "G semisimple" to the assumptions of Theorem 2.2. | I'm going to add the condition "G semisimple" to the assumptions of Theorem 2.2. | ||
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+ | Nachtrag: Now that the statement has changed to include compact Lie groups only: wouldn't it be clearer to state that compact Lie groups are isomorphic to subgroups of O(n) not just of GL(n,R)? | ||
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+ | Or perhaps one should divide the theorem into two parts: | ||
+ | a) each semisimple Lie group is isomorphic to a subgroup of GL(n,R) | ||
+ | b) each compact. Lie group is isomorphic to a subgroup of O(n) |
Latest revision as of 11:59, 9 June 2010
Theorem 2.2 is wrong as stated. A counterexample is given on page 83 of [Carter&Segal&Macdonald1995].
For each Lie group one has the adjoint representation $Ad:G\rightarrow GL\left(g\right)$. This representation is faithful if $G$ is semisimple. Thus Theorem 2.2 is correct for semisimple Lie groups.
I'm going to add the condition "G semisimple" to the assumptions of Theorem 2.2.
Nachtrag: Now that the statement has changed to include compact Lie groups only: wouldn't it be clearer to state that compact Lie groups are isomorphic to subgroups of O(n) not just of GL(n,R)?
Or perhaps one should divide the theorem into two parts: a) each semisimple Lie group is isomorphic to a subgroup of GL(n,R) b) each compact. Lie group is isomorphic to a subgroup of O(n)