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## Tuesday, May 29, 2007

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Mapping the geometry of the F4 group

While browsing the arxiv, I came upon a cool new paper on the group F4. The paper contains a plethora of mathematical treasures, including: the exceptional Jordan algebra, OP^2 and the generalized F4 Euler angle construction I mentioned to Kea and Carl earlier this year. Pierre Ramond explored these issues in the context of M-theory back in 1998 (hep-th/9808190). More recently, the physical context has been extended to extremal black holes in N=2 Maxwell-Einstein supergravities (hep-th/0512296).

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.

While browsing the arxiv, I came upon a cool new paper on the group F4. The paper contains a plethora of mathematical treasures, including: the exceptional Jordan algebra, OP^2 and the generalized F4 Euler angle construction I mentioned to Kea and Carl earlier this year. Pierre Ramond explored these issues in the context of M-theory back in 1998 (hep-th/9808190). More recently, the physical context has been extended to extremal black holes in N=2 Maxwell-Einstein supergravities (hep-th/0512296).

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.

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## 3 comments:

Oooo. Nice resource. Thanks for the link.

Hi Kneemo,

Thanks for these F4 [and E6], SO(9) and BPS references.

I am a bit confused when comparing the 'Mapping the geometry of the F4 group' paper to the paper by Gannon 'Monstrous Moonshine: The first twenty-five years'

http://lanl.arxiv.org/abs/math/0402345

Gannon mentions, after Figure 2 on page 7, that a two-folding of E7 yields F4 and is related to the Baby Monster.

A triple-folding of E6 yields G2 related to the Monster and a Fisher group.

I wonder why Gannon did not mention that F4 might be related to both E7 and E6.

It is certainly possible that I may overlooked that he did.

To relate F4 to E7 in a nice way, one uses a Freudenthal triple system (FTS), which contains the exceptional Jordan algebra and its dual, plus two real coordinates. The automorphism group of the exceptional Jordan algebra is F4 while the automorphism group of the FTS is a real, non-compact form of E7.

It's possible that Gannon isn't yet familiar with the FTS construction.

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