Mapping the geometry of the F4 group

Abstract

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 × 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 OP 2 = F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.

Mapping the geometry of the E6 group

“…As argued in [1] we expect to find a 28-dimensional subgroup of F 4 ,which, acting by adjunction, defines an automorphism of V . This can be done by noticing that V commutes with the first 28 matrices c i , i = 1, .…”

“…This is particularly interesting, because recently it has been argued that this group could be the most promising for unification in GUT theories [3,6]. To parameterize the group we have used the generalized Euler angles method, a technique we introduced in [2,1] to give the most simple expression for the invariant measure on the group, while at the same time still being able to provide an explicit expression for the range of the parameters. Both these requirements are necessary in order to minimize the computation power needed for computer simulations, for example, of lattice models.…”

“…, 28, which generate an SO(8) subgroup of F 4 . We found convenient to introduce the change of basẽ and the tilded matrices are the ones introduced in [1].…”

“…However for large dimensions it becomes quite hard to find a realization, which is able at the same time to provide both a reasonably simple form for the measure and an explicit determination for the range of the angles. In this paper we solve this problem for the exceptional Lie group E 6 , by constructing a generalization of the Euler parametrization, with a technique we have introduced in [2] and fully developed in [1]. In section 2 we explain how the e 6 algebra can be represented using a theorem due to Chevalley and Schafer.…”

“…The exceptional Jordan algebra is 27 dimensional, just like the principal fundamental representation of e 6 . In our previous paper [1] we determined the algebra of derivations D and used it to obtain an irreducible representation of the exceptional algebra f 4 . To obtain a representation of e 6 we only need to determine the matrix representation of R Y .…”