PlanetMath: one-parameter subgroup

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one-parameter subgroup

(Definition)

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

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

that is also a differentiable map at the same time. We view $\mathbb{R}$ additively and

multiplicatively, so that $\phi(r+s)=\phi(r)\phi(s)$ .

Examples.

If $G=\operatorname{GL}(n,k)$ , where $k=\mathbb{R}$ or $\mathbb{C}$ , then any one-parameter subgroup has the form
$\displaystyle \phi(t)=e^{tA},$
where $A=\frac{d\phi}{dt}(0)$ is an $n\times n$ matrix over . The matrix 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.
If $G=\operatorname{O}(n,\mathbb{R})\subseteq\operatorname{GL}(n,\mathbb{R})$ , the orthogonal group over , then any one-parameter subgroup has the same form as in the example above, except that is skew-symmetric: $A^{\operatorname{T}}=-A$ .
If $G=\operatorname{SL}(n,\mathbb{R})\subseteq\operatorname{GL}(n,\mathbb{R})$ , the special linear group over , 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.
If $G=\operatorname{U}(n)=\operatorname{O}(n,\mathbb{C})\subseteq\operatorname{GL}(n,\mathbb{C})$ , the unitary group over , then any one-parameter subgroup has the same form as in the example above, except that is skew-Hermitian: $A=-A^{*}=-\overline{A}^{\operatorname{T}}$ and $\operatorname{tr}(A)=0$ .

"one-parameter subgroup" is owned by CWoo. [ full author list (2) ]

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Other names:

1-parameter subgroup

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Cross-references: unitary group, operator, trace, special linear group, skew-symmetric, orthogonal group, matrix exponential, map, exponential, Lie algebra, identity, one-to-one correspondence, property, tangent vector, matrix, differentiable map, group homomorphism, Lie group
There are 4 references to this entry.

This is version 4 of one-parameter subgroup, born on 2004-12-15, modified 2005-06-13.
Object id is 6583, canonical name is OneParameterSubgroup.
Accessed 3275 times total.

Classification:

AMS MSC:	22E10 (Topological groups, Lie groups :: Lie groups :: General properties and structure of complex Lie groups)
	22E15 (Topological groups, Lie groups :: Lie groups :: General properties and structure of real Lie groups)

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