On the Euler angles for SU(N)

TL;DR

This paper revisits the Euler parametrization of SU(N) groups, deriving the invariant measure and parameter ranges, and interprets the parameters as generalized Euler angles through topological and geometrical analysis.

Contribution

It provides a detailed construction of the Euler parametrization for SU(N), including the invariant measure and parameter ranges, and interprets the parameters as generalized Euler angles.

Findings

Derived the invariant measure on SU(N)

Determined the full range of parameters for the Euler angles

Showed SU(N+1) as a fibration of U(N) over complex projective space

Abstract

In this paper we reconsider the problem of the Euler parametrization for the unitary groups. After constructing the generic group element in terms of generalized angles, we compute the invariant measure on SU(N) and then we determine the full range of the parameters, using both topological and geometrical methods. In particular, we show that the given parametrization realizes the group $SU(N+1)$ as a fibration of U(N) over the complex projective space $\mathbb{CP}^n$. This justifies the interpretation of the parameters as generalized Euler angles.