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I have known that $sl(n,C)$ is a simple Lie algebra and it is a linear representation of $SL(n,C)$ by adjoint.

But is it an irreducible representation of $SL(n,C)$ and is there some relation between this and its property of being simple?

wsh
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    See https://math.stackexchange.com/questions/3188894/adjoint-representation-is-irreducible-iff-mathfrakg-is-simple – Kyle Miller Jan 30 '23 at 17:40
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    That "duplicate" concerns the adjoint representation of the Lie algebra. This question seems to be about a representation of the group. Granted there's a strong connection, but it's not identical in my eyes. – Torsten Schoeneberg Jan 30 '23 at 20:46
  • wsh, can you clarify whether you mean the representation of the group, or its differential for the Lie algebra? We have $ {\displaystyle d_{e}\operatorname {Ad} =\operatorname {ad} } $. – Dietrich Burde Jan 31 '23 at 15:44
  • the representation of the group – wsh Feb 01 '23 at 11:39

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