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Does it have any physical meaning

  • It is so badly written, its hard to tell what are you asking: Are you asking if every Lie group embeds in the automorphism group of its Lie algebra? Or that it is isomorphic to the automorphism group of its Lie algebra? Are you asking if this is true for each finite-dimensional unitary group (I assume this is what $U$ stands for)? Vote to close for now. – Moishe Kohan Aug 16 '17 at 06:08

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A Lie group acts on its Lie algebra via the "adjoint representation". This action need not be faithful: in $\text{SL}_2(\Bbb R)$, $-I$ acts trivially on the Lie algebra.

Angina Seng
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  • do you mean lie algebra acts on dual space of the vector space on which its lie group acts on ? Or do you mean that the Lie algebra forms the dual of the vector space ? –  Aug 15 '17 at 07:02
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    @VidyeshRaoA There was no dual mentioned here. The Lie group acts on the Lie algebra. There is no "the" space on which either acts, as they each have many representations. – Tobias Kildetoft Aug 15 '17 at 08:11