3 Matching Results

Search Results

Advanced search parameters have been applied.

Mapping the geometry of the E6 group

Description: In this paper we present a construction for the compact form of the exceptional Lie group E{sub 6} by exponentiating the corresponding Lie algebra e{sub 6}, which we realize as the sum of f{sub 4}, the derivations of the exceptional Jordan algebra J{sub 3} of dimension 3 with octonionic entries, and the right multiplication by the elements of J{sub 3} with vanishing trace. Our parameterization is a generalization of the Euler angles for SU(2) and it is based on the fibration of E{sub 6} via a F{sub 4} subgroup as the fiber. It makes use of a similar construction we have performed in a previous article for F{sub 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{sub 6} group manifold.
Date: October 1, 2007
Creator: Cerchiai , Bianca; Bernardoni, Fabio; Cacciatori, Sergio L.; Cerchiai, Bianca L. & Scotti, Antonio
Partner: UNT Libraries Government Documents Department

Mapping the geometry of the F4 group

Description: In this paper, we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 Hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2 = F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.
Date: May 28, 2007
Creator: Bernardoni, Fabio; Cacciatori, Sergio L; Scotti, Antonio & Cerchiai, Bianca L.
Partner: UNT Libraries Government Documents Department

Euler angles for G2

Description: We provide a simple parameterization for the group G2, which is analogous to the Euler parameterization for SU(2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions. In particular this allows us to obtain a simple expression for the Haar measure on G2. Moreover, as a by-product it yields a concrete realization and an Einstein metric for H.
Date: March 10, 2005
Creator: Cacciatori, Sergio; Cerchiai, Bianca Letizia; della Vedova,Alberto; Ortenzi, Giovanni & Scotti, Antonio
Partner: UNT Libraries Government Documents Department