I'm currently learning Lie groups/algebras and I am trying to find the infinitesimal generators of the special orthogonal group SO(n). It is the n-dimensions that throws me off. I know that the answer is [itex]n(n-1)/2[/itex] generators of the form,(adsbygoogle = window.adsbygoogle || []).push({});

[tex]X_{\rho,\sigma}=-i\left(x_{\rho}\frac{\partial}{\partial x_{\sigma}}-x_{\sigma}\frac{\partial}{\partial x_{\rho}}\right)[/tex]

where [itex]1\leq x_{\rho}\leq n[/itex] such that [itex]\sigma > \rho[/itex], but how do I get this? The only way I could think of is somehow trying to find the n-dimensional rotation matrix in general and then going from there (I have actually tried this and it gets me something similar)--there must be a simpler way though.

I also run into this problem in other areas of the book I am using. For example attempting to establish that the Lie algebra sl(n,C) is an ideal of gl(n,C). I guess I am looking for some advice on how to compute these algebras in n-dimensions. Thanks in advance for any help/suggestions, and let me know if I can make anything more clear.

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# Question on N-dimensional Lie Groups

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