The volume of a compact Lie group
The volume of a compact Lie group
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DOI：
10.1007/bf01392542
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发表时间：
1980-02
影响因子：
3.1
通讯作者：
I. G. MacDonald
中科院分区：
文献类型：
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作者：
I. G. MacDonald
I.设G是一个紧李群，q是它的李代数，认为是G在单位元e处的切空间。在向量空间g上选择一个勒贝格测度2。借助于G在e处的图，我们可以从2构造G中e的邻域上的一个不变测度，然后我们可以通过在G中的平移将其扩展为G上的一个Haar测度/t。本文的目的是建立~（G）的公式（（1）），即G相对于测度/~的体积，它是2的函数。首先，从适当选择的Chevalley基的复化g，我们可以构建一个“整数格”gz，这是一个格在。q和Z上的李代数。由于记法不当，令2（g/gz）表示gz的基本平行六面体的体积（相对于2），单位为g。其次，众所周知，除了挠率之外，流形G具有与奇维球面的乘积相同的上同调，比如维数
I. Let G be a compact Lie group and let, q be its Lie algebra, thought of as the tangent space to G at the identity element e. Choose a Lebesgue measure 2 on the vector space g. By means of a chart of G at e we can construct from 2 a translation-invariant measure on a neighbourhood of e in G, and then we can extend this by translation in G to a Haar measure/t on G. The purpose of this note is to establish a formula ((1) below) for~(G), the volume of G relative to the measure/~, as a function of 2.There are two ingredients in the formula. Firstly, from a suitably chosen Chevalley basis of the complexification of g we can construct an" integer lattice" gz, which is a lattice in. q and a Lie algebra over Z. By abuse of notation, let 2 (g/gz) denote the volume (with respect to 2) of a fundamental parallelepiped for gz in g. Secondly, it is well-known that the manifold G has the same cohomology, apart from torsion, as a product of odd-dimensional spheres, say of dimensions