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Title: 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 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.

Authors:
; ; ;
Publication Date:
Research Org.:
Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
Physics Division
OSTI Identifier:
948502
Report Number(s):
LBNL-1517E
TRN: US0901666
DOE Contract Number:
DE-AC02-05CH11231
Resource Type:
Journal Article
Resource Relation:
Journal Name: Advances in Theoretical and Mathematical Physics; Journal Volume: 12; Journal Issue: 4; Related Information: Journal Publication Date: 2008
Country of Publication:
United States
Language:
English
Subject:
72; 97; ALGEBRA; CONSTRUCTION; GEOMETRY; JORDAN; LIE GROUPS; MATRICES; exceptional Lie groups, Haar measure, coset spaces, geometry

Citation Formats

Bernardoni, Fabio, Cacciatori, Sergio L, Scotti, Antonio, and Cerchiai, Bianca L. Mapping the geometry of the F4 group. United States: N. p., 2007. Web.
Bernardoni, Fabio, Cacciatori, Sergio L, Scotti, Antonio, & Cerchiai, Bianca L. Mapping the geometry of the F4 group. United States.
Bernardoni, Fabio, Cacciatori, Sergio L, Scotti, Antonio, and Cerchiai, Bianca L. Mon . "Mapping the geometry of the F4 group". United States. doi:. https://www.osti.gov/servlets/purl/948502.
@article{osti_948502,
title = {Mapping the geometry of the F4 group},
author = {Bernardoni, Fabio and Cacciatori, Sergio L and Scotti, Antonio and Cerchiai, Bianca L.},
abstractNote = {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.},
doi = {},
journal = {Advances in Theoretical and Mathematical Physics},
number = 4,
volume = 12,
place = {United States},
year = {Mon May 28 00:00:00 EDT 2007},
month = {Mon May 28 00:00:00 EDT 2007}
}
  • 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 wemore » 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.« less
  • 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 parametrization is a generalization of the Euler angles for SU(2) and it is based on the fibration of E{sub 6} via an F{sub 4} subgroup as the fiber. It makes use of a similar construction wemore » 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.« less
  • No abstract prepared.
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  • The periplasmic chaperone FaeE of E. coli F4 fimbriae crystallizes in three crystal forms. F4 (formerly K88) fimbriae from enterotoxigenic Escherichia coli are assembled via the FaeE/FaeD chaperone/usher pathway. The chaperone FaeE crystallizes in three crystal forms, all belonging to space group C2. Crystals of form 1 diffract to 2.3 Å and have unit-cell parameters a = 195.7, b = 78.5, c = 184.6 Å, β = 102.2°. X-ray data for crystal form 2 were collected to 2.7 Å using an SeMet variant of FaeE. The crystals have unit-cell parameters a = 136.4, b = 75.7, c = 69.4 Å,more » β = 92.8°. Crystals of form 3 were formed in a solution containing the FaeE–FaeG complex and diffract to 2.8 Å. Unit-cell parameters are a = 109.7, b = 78.6, c = 87.8 Å, β = 96.4°.« less