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Consider a unitary group $G$ which is a proper subgroup of $U(N)$ and which is generated by an algebra $\{h_i\}$. If I now write the set of all matrices $Z=\sum_{i}z_{i}h_{i}$, where $z_{i}\in\mathbb{C}^1$, then we have an algebra which is a subalgebra of $SL(N,\mathbb{C})$. I want to know the name of this subalgebra in general..

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In physics there is a tendency to conflate a Lie group and its Lie algebra. I've fixed up some missing braces. –  David Roberts Sep 18 '12 at 7:15
That makes more sense now David, but still not perfect sense. Should "generated by an algebra $\lbrace h_i\rbrace$" be "generated as an algebra by $\lbrace h_i \rbrace$"? –  Mark Grant Sep 18 '12 at 8:07
Ah, thanks David, that makes sense now. [Have deleted earlier comment] –  Yemon Choi Sep 18 '12 at 8:57
I am not sure to correctly understand but I have the impress that it is just the description of the complexification of a Lie subalgebra of u(n). The complexification lives in general in gl(n)(C) (not sl(n)(C)) –  user25309 Sep 18 '12 at 9:55

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