Mathematics Colloquia and Seminars

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We provide simple parametrizations for the exceptional Lie groups G2, F4 and E6, based on their fibration structure. For the compact case, we construct a realization which is a generalization of the Euler angles for SU(2), while for the non compact version of G2(2)/SO(4) we compute the Iwasawa decomposition. This allows us to obtain not only an explicit expression for the Haar measure on the group manifold, but also for the cosets G2/SO(4), F4/Spin(9), E6/F4 and G2,(2)/SO(4) that we used to find the concrete realization of the general element of the group. Moreover, as a by-product, in the simplest cases of G2/SO(4), we have been able to compute an Einstein metric and the vielbein for these cosets. These cosets are relevant in physics, e.g. for supergravity theories and the black hole attractor mechanism.