Talk:Lie groups I: Definition and examples

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(Created page with 'Theorem 2.2 is wrong as stated. A counterexample is given on page 83 of {{cite|Carter&Segal&Macdonald1995}}. For each Lie group one has the adjoint representation $Ad:G\rightarr…')
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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:
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a) each semisimple Lie group is isomorphic to a subgroup of GL(n,R)
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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)

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