Self maps of Lie groups

李群的自映射

基本信息

批准号：
10640087
负责人：
OSHIMA Hideaki
金额：
$ 0.64万
依托单位：
IBARAKI UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640087/
关键词：
Lie group self map homotopy nilpotency class Hopf space rank Samelson product commutator 冪零指数

项目摘要

1. Let X be a connected CW Hopf space with a multiplication μ. For a pathconnected and pointed space A, the homotopy set [A, X] of continuous maps from A to X has a binary operation "+" induced from μ. We write ([A, X], +) = [A, X ; μ] which is an algebraic loop. Our first result is the following : If A and X are connected CW Hopf spaces with at most three cells, then [A, X ; μ] becomes a group and its group structure can be determined. By the way, there are fifteen connected CW Hopf spaces with at most three cells and they have in general many multiplications. Therefore there were many groups we should compute.2. Let G be a connected Lie group and μィイD20ィエD2 the multiplication of G. Then the algebraic loop [A, G ; μィイD20ィエD2] is a group and it satisfies the relation : nil[A, G ; μィイD20ィエD2] 【less than or equal】 cat(A) as proved by G. W. Whitehead, where nil denotes the nilpotency class and cat denotes the Lusternik-Schnirelmann category with cat(*) = 0. We are interested in estimation of nil[A, G ; μィイD20ィエD2] from below. In the first place, though it is the most interested case, we have consider the case A = G. We have two conjectures :(1) If G is simple, then nil[G, G ; μィイD20ィエD2] 【greater than or equal】 rank(G).(2) If G is simple and rank(G) 【greater than or equal】 2, then nil[G, G ; μィイD20ィエD2] 【greater than or equal】 2.Of course if (1) is affirmative then so is (2). Without the assumption "simple", two conjectures are in general false. We proved (1) affirmative when G is one of SO(3), SU(3), SU(4), Sp(2), Spin(7) and GィイD22ィエD2 ; and (2) affirmative when G is one of SU(5), SU(6), Sp(3), Spin(8), EィイD26ィエD2, EィイD28ィエD2, and FィイD24ィエD2. We determined the group [G, G ; μィイD20ィエD2] completely for G = SO(3), SU(3), Sp(2) and almost completely for G = GィイD22ィエD2. We have several fragmental results for G not simply connected or not simple.

1. 令 X 为具有乘法 μ 的连通 CW Hopf 空间对于路径连通的有向空间 A，从 A 到 X 的连续映射的同伦集 [A, X] 具有由 μ We 导出的二元运算“+”。写 ([A, X], +) = [A, X; μ] 这是一个代数循环：如果 A 和 X ； μ] 成为一个群，并且可以确定其群结构。顺便说一句，有 15 个连通的 CW Hopf 空间，最多有 3 个单元，并且它们通常有很多乘法，因此我们需要计算很多群。 2．是一个连通的李群，μIID20ieD2 是 G 的乘法。则代数环 [A, G; μIIID20ieD2] 是一个群，并且满足关系： nil[A, G ; μIID20D2] [小于或等于] cat(A) 由 G. W. Whitehead 证明，其中 nil 表示幂零类，cat 表示 cat(*) = 0 的 Lusternik-Schnirelmann 类别。我们对 nil[ 的估计感兴趣A, G ; μIID20ieD2] 首先，虽然这是最感兴趣的情况，但我们考虑了 A = G 的情况。两个猜想：(1) 如果 G 是简单的，则 nil[G, G ; μiiD20ieD2] 【大于或等于】rank(G)。(2) 如果 G 是简单的，并且rank(G) 【大于或等于】 2 ，则 nil[G, G ; μIIiD20ieD2] 【大于或等于】 2.当然，如果 (1) 成立，则 (2) 也成立。 “简单”，当 G 是 SO(3)、SU(3)、SU(4)、Sp(2)、Spin(7) 和 G D22 D2 之一时，我们证明 (1) 是肯定的。；并且(2)当G是SU(5)、SU(6)、Sp(3)、Spin(8)、E D26 D2、E D28 D2 之一时是肯定的，并且F D24 D2。对于 G = SO(3)、SU(3)、Sp(2)，我们完全确定了群 [G, G ; μ D20 D2]；对于 G = G D22 D2，我们几乎完全确定了群 [G, G ; μ D20 D2]。对于 G 不是简单连接或不简单。