Properties
The following table summarizes the properties of the various maps mentioned in the definition
| Lie group homomorphism:
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Lie group automorphism:
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| Lie group homomorphism:
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Lie algebra automorphism:
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Lie algebra homomorphism:
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Lie algebra derivation:
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The image of G under the adjoint representation is denoted by AdG. If G is connected, the kernel of the adjoint representation coincides with the kernel of Ψ which is just the center of G. Therefore the adjoint representation of a connected Lie group G is faithful if and only if G is centerless. More generally, if G is not connected, then the kernel of the adjoint map is the centralizer of the identity component G0 of G. By the first isomorphism theorem we have
Read more about this topic: Adjoint Representation Of A Lie Group
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