that is also a differentiable map at the same time. We view additively and multiplicatively, so that .
If , where or , then any one-parameter subgroup has the form
where is an matrix over . The matrix is just a tangent vector to the Lie group . This property establishes the fact that there is a one-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 (the Lie algebra) and one-parameter subgroups is established via the exponential map instead of the matrix exponential.
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.
If , the unitary group 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 .
|Date of creation||2013-03-22 14:54:01|
|Last modified on||2013-03-22 14:54:01|
|Last modified by||CWoo (3771)|