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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