One-parameter subgroup
of a Lie group over a normed field
An analytic homomorphism of the additive group of the field into
, that is, an analytic mapping
such that
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The image of this homomorphism, which is a subgroup of , is also called a one-parameter subgroup. If
, then the continuity of the homomorphism
implies that it is analytic. If
or
, then for any tangent vector
to
at the point
there exists a unique one-parameter subgroup
having
as its tangent vector at the point
. Here
,
, where
is the exponential mapping. In particular, any one-parameter subgroup of the general linear group
has the form
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If is a real Lie group endowed with a two-sidedly invariant pseudo-Riemannian metric or affine connection, then the one-parameter subgroups of
are the geodesics passing through the identity
.
References
[1] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) |
[2] | J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) |
[3] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) |
Comments
References
[a1] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) |
[a2] | N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1972) pp. Chapt. 2; 3 |
[a3] | G. Hochschild, "Structure of Lie groups" , Holden-Day (1965) |
One-parameter subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-parameter_subgroup&oldid=11235