Mapping the geometry of the E6 group

UCB-PTH-07/14 IFIC/07−41 FTUV/07−MMDD Mapping the geometry of the E 6 group arXiv:0710.0356v1 [math-ph] 1 Oct 2007 Fabio Bernardoni 1∗ , Sergio L. Cacciatori 2† , Bianca L. Cerchiai 3‡ and Antonio Scotti 4§ Departament de F´ isica Te` rica, IFIC, Universitat de Val` ncia - CSIC o e Apt. Correus 22085, E-46071 Val` ncia, Spain. e Dipartimento di Scienze Fisiche e Matematiche, Universit` dell’Insubria, a Via Valleggio 11, I-22100 Como. Lawrence Berkeley National Laboratory Theory Group, Bldg 50A5104 1 Cyclotron Rd, Berkeley CA 94720 USA Dipartimento di Matematica dell’Universit` di Milano, a Via Saldini 50, I-20133 Milano, Italy. INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano. Abstract In this paper we present a construction for the compact form of the exceptional Lie group E 6 by expo- nentiating the corresponding Lie algebra e 6 , which we realize as the the sum of f 4 , the derivations of the exceptional Jordan algebra J 3 of dimension 3 with octonionic entries, and the right multiplication by the elements of J 3 with vanishing trace. Our parametrization is a generalization of the Euler angles for SU (2) and it is based on the fibration of E 6 via a F 4 subgroup as the fiber. It makes use of a similar construction we have performed in a previous article for F 4 . An interesting first application of these results lies in the fact that we are able to determine an explicit expression for the Haar invariant measure on the E 6 group manifold. ∗ Fabio.Bernardoni@ific.uv.es † sergio.cacciatori@uninsubria.it ‡ BLCerchiai@lbl.gov § antonio.scotti@gmail.com