# Lie Algebra of U(N) and SO(N)

U(N) and SO(N) are quite important groups in physics. I thought I would find this with an easy google search. Apparently NOT! What is the Lie algebra and Lie bracket of the two groups?

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If you have a copy of John Lee's book on Smooth manifolds, you can find the important details of these computations in chapter 8. I don't have my book with me right now, but I'll update with specific page information tomorrow. – jmracek Oct 9 '12 at 3:55

The Lie algebra for $U(N)$ consists of $N\times N$ skew-Hermitian matrices, and the Lie algebra for $SO(N)$ consists of $N\times N$ skew-symmetric matrices. In both cases, the Lie bracket is given by the ordinary commutator $[A,B] = AB-BA$.
You may wish to check check that for $U(2)$, this gives the familiar definition in terms of angular momentum.