## You are here

Homeone-parameter subgroup

## Primary tabs

# 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. 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. 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. 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. 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: $A=-A^{{*}}=-\overline{A}^{{\operatorname{T}}}$ and $\operatorname{tr}(A)=0$.

## Mathematics Subject Classification

22E15*no label found*22E10

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections