请输入您要查询的字词:

 

单词 OneparameterSubgroup
释义

one-parameter subgroup


Let G be a Lie GroupMathworldPlanetmath. Aone-parameter subgroup of G is a group homomorphismMathworldPlanetmath

ϕ:G

that is also a differentiablemap at the same time. We view additively and Gmultiplicatively, so that ϕ(r+s)=ϕ(r)ϕ(s).

Examples.

  1. 1.

    If G=GL(n,k), where k= or , then any one-parameter subgroup has the form

    ϕ(t)=etA,

    where A=dϕdt(0) is an n×n matrix over k. The matrix A is just atangent vector to the Lie group GL(n,k). This property establishes the fact that there is aone-to-one correspondence between one-parameter subgroups and tangent vectors of GL(n,k). The same relationship holds for a general Lie group.The one-to-one correspondence between tangent vectors at the identityPlanetmathPlanetmath (theLie algebra) and one-parameter subgroups is established via the exponentialmap instead of the matrix exponentialMathworldPlanetmath.

  2. 2.

    If G=O(n,)GL(n,), the orthogonal groupMathworldPlanetmath over R, thenany one-parameter subgroup has the same form as in the example above, except that A is skew-symmetric:AT=-A.

  3. 3.

    If G=SL(n,)GL(n,), the special linear groupMathworldPlanetmath over R,then any one-parameter subgroup has the same form as in the example above, except that tr(A)=0, wheretr is the trace operator.

  4. 4.

    If G=U(n)=O(n,)GL(n,), the unitarygroupMathworldPlanetmath over C, then any one-parameter subgroup has the same form as in the example above, except that A is skew-Hermitian (http://planetmath.org/SkewHermitianMatrix): A=-A*=-A¯T and tr(A)=0.

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 21:20:11