Mapping the geometry of the F4 group

Details Mapping the geometry of the F4 group Mapping the geometry of the F4 group

2007-05-28

arxiv.org/abs/0705.3978v2

Adv.Theor.Math.Phys.12:889-994,2008

Physics

Mathematical Physics

50 pages; some typos corrected

Scientific paper

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.

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