# Representing elements of U(N) or SO(N) by elementary rotations exp(i phi_n sigma_n)

Given a (orthogonal) basis $(\sigma_n)_{n=1,\dots,K}$ of the algebra $u(N)$, we can represent any element $U$ of the corresponding group $U(N)$ in the form

$U=e^{i\sum_{n=1}^K\varphi_n\sigma_n}$.

Is it also possible to represent every element of $U(N)$ in the following form as a product of elementary (Givens) rotations

$U=\prod_{n=1}^K e^{i\alpha_n\sigma_n}$ ?

Is the corresponding property given for the special orthogonal groups $SO(N)$? For SO(3) it is -- in that case, $\alpha_i$ are the well-known Euler angles.

thanks a lot in advance, Robert

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## 1 Answer

See this paper.

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arxiv.org/abs/math-ph/0205016 generalized Euler angles for u n –  Alexander Chervov May 10 '12 at 17:37
it is cited in the reference I gave. –  Abdelmalek Abdesselam May 10 '12 at 17:41