Faithfulness of adjoint representation of Lie algberas

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Are there any simple or useful conditions (necessary & sufficient) under which the adjoint representation lie algebra is faithful ? One sufficient condition is semisimplicity, but perhaps this is not necessary.

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1 Answer

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The adjoint representation of a Lie algebra has a kernel which is very easy to describe: It is the center of the Lie algebra (this is easy to check directly from the definition).

So the adjoint representation is faithful iff the Lie algebra has a trivial center.

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