one-parameter subgroup
Let be a Lie Group. Aone-parameter subgroup of is a group homomorphism
that is also a differentiablemap at the same time. We view additively and multiplicatively, so that .
Examples.
- 1.
If , where or , then any one-parameter subgroup has the form
where is an matrix over . The matrix is just atangent vector to the Lie group . This property establishes the fact that there is aone-to-one correspondence between one-parameter subgroups and tangent vectors of . The same relationship holds for a general Lie group.The one-to-one correspondence between tangent vectors at the identity
(theLie algebra) and one-parameter subgroups is established via the exponentialmap instead of the matrix exponential
.
- 2.
If , the orthogonal group
over , thenany one-parameter subgroup has the same form as in the example above, except that is skew-symmetric:.
- 3.
If , the special linear group
over ,then any one-parameter subgroup has the same form as in the example above, except that , where is the trace operator.
- 4.
If , the unitarygroup
over , then any one-parameter subgroup has the same form as in the example above, except that is skew-Hermitian (http://planetmath.org/SkewHermitianMatrix): and .