one-parameter subgroup

Let $G$ be a Lie Group. A one-parameter subgroup of $G$ is a group homomorphism

$$\varphi :ℝ\to G$$

that is also a differentiable map at the same time. We view $ℝ$ additively and $G$ multiplicatively, so that $\varphi (r+s)=\varphi (r)\varphi (s)$.

Examples.

1. If $G=\mathrm{GL}(n,k)$, where $k=ℝ$ or $ℂ$, then any one-parameter subgroup has the form

   $$\varphi (t)=e^{tA},$$

   where $A=\frac{d\varphi }{dt}(0)$ is an $n\times n$ matrix over $k$. The matrix $A$ is just a tangent vector to the Lie group $\mathrm{GL}(n,k)$. This property establishes the fact that there is a one-to-one correspondence between one-parameter subgroups and tangent vectors of $\mathrm{GL}(n,k)$. The same relationship holds for a general Lie group. The one-to-one correspondence between tangent vectors at the identity (the Lie algebra) and one-parameter subgroups is established via the exponential map instead of the matrix exponential.

2. If $G=\mathrm{O}(n,ℝ)\subseteq \mathrm{GL}(n,ℝ)$, the orthogonal group over $R$, then any one-parameter subgroup has the same form as in the example above, except that $A$ is skew-symmetric: $A^{\mathrm{T}}=-A$.

3. If $G=\mathrm{SL}(n,ℝ)\subseteq \mathrm{GL}(n,ℝ)$, the special linear group over $R$, then any one-parameter subgroup has the same form as in the example above, except that $\mathrm{tr}(A)=0$, where $\mathrm{tr}$ is the trace operator.

4. If $G=\mathrm{U}(n)=\mathrm{O}(n,ℂ)\subseteq \mathrm{GL}(n,ℂ)$, the unitary group 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): $A=-A^{*}=-{\overline{A}}^{\mathrm{T}}$ and $\mathrm{tr}(A)=0$.

Title	one-parameter subgroup
Canonical name	OneparameterSubgroup
Date of creation	2013-03-22 14:54:01
Last modified on	2013-03-22 14:54:01
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 22E15
Classification	msc 22E10
Synonym	1-parameter subgroup