映射F4组的几何结构
Mapping the geometry of the F4 group
摘要：
本文通过对相应的李代数f4进行指数化，给出了例外李群F4的紧形式的构造。我们把F4看作例外Jordan代数的自同构群，它的元素是3 × 3的八元厄米矩阵.我们使用一个参数化，它概括了SU（2）的欧拉角，并基于F4通过Spin（9）子群作为纤维的纤维化。这种技术使我们能够确定一个明确的表达的Haar不变测度的F4群流形。这些结果除了揭示F4及其陪集流形OP 2 =F4/Spin（9）（八元数射影平面）的结构外，还为研究E6（F4是E6的（极大）子群）提供了必要条件。
英文摘要：
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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