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This is more of a maths question, however I'm confident there are some hardcore mathematical physicists out there amongst you. It's more of a curiosity, and I'm not sure how to address it to convince myself of an answer.

I have a Lie group homomorphism [itex] \rho : G \rightarrow GL(n, \mathbb{C}) \hspace{0.5cm}(1) [/itex]

and a Lie Algebra homomorphism [itex] \hat{\rho} : g \rightarrow gl(n, \mathbb{C}) \hspace{1cm}(2) [/itex]

which are the group and algebra representations on the space of nxn matrices viewed as a vector space.

Now, the difference in the images of these two maps is that in (1) it has to be a group, so formally it is defined as the group of Automorphisms on a vector space, Aut(V), whereas in (2) it has to be an algebra, and I've read somewhere this is defined formally as End(W), and seen it written as "the Lie algebra of Endomorphisms of a vector space W".

Two questions:

- How does one properly define the Lie Algebra representation? I'm not quite sure I understand what is going on with the Endomorphisms being an algebra?

- If, in my original maps above, g is the Lie algebra of G, and I rewrite the maps as

[itex] \rho : G \rightarrow Aut(V) \hspace{0.5cm}(3) [/itex]

[itex] \hat{\rho} : g \rightarrow End(W) \hspace{1cm}(4) [/itex]

What situation am I representing if I choose V=W ? Does demanding V=W output the "analogous" representation, say the fundamental of the group and the fundamental of the algebra?

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# Lie Group v Lie algebra representation

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