# one-parameter subgroup

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

 $\phi\colon\mathbb{R}\to G$

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

Examples.

1. 1.

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

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

where $A=\frac{d\phi}{dt}(0)$ is an $n\times n$ matrix over $k$. The matrix $A$ is just a tangent vector to the Lie group $\operatorname{GL}(n,k)$. This property establishes the fact that there is a one-to-one correspondence between one-parameter subgroups and tangent vectors of $\operatorname{GL}(n,k)$. 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. 2.

If $G=\operatorname{O}(n,\mathbb{R})\subseteq\operatorname{GL}(n,\mathbb{R})$, 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: $A^{\operatorname{T}}=-A$.

3. 3.

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

4. 4.

If $G=\operatorname{U}(n)=\operatorname{O}(n,\mathbb{C})\subseteq\operatorname{GL}% (n,\mathbb{C})$, 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): $A=-A^{*}=-\overline{A}^{\operatorname{T}}$ and $\operatorname{tr}(A)=0$.

Title one-parameter subgroup OneparameterSubgroup 2013-03-22 14:54:01 2013-03-22 14:54:01 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 22E15 msc 22E10 1-parameter subgroup