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One-parameter subgroup

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of a Lie group $ G $ over a normed field $ K $

An analytic homomorphism of the additive group of the field $ K $ into $ G $, that is, an analytic mapping $ \alpha : K \rightarrow G $ such that

$$ \alpha ( s + t) = \alpha ( s) \alpha ( t),\ s, t \in K. $$

The image of this homomorphism, which is a subgroup of $ G $, is also called a one-parameter subgroup. If $ K = \mathbf R $, then the continuity of the homomorphism $ \alpha : K \rightarrow G $ implies that it is analytic. If $ K = \mathbf R $ or $ \mathbf C $, then for any tangent vector $ X \in T _ {e} G $ to $ G $ at the point $ e $ there exists a unique one-parameter subgroup $ \alpha : K \rightarrow G $ having $ X $ as its tangent vector at the point $ t = 0 $. Here $ \alpha ( t) = \mathop{\rm exp} tX $, $ t \in K $, where $ \mathop{\rm exp} : T _ {e} G \rightarrow G $ is the exponential mapping. In particular, any one-parameter subgroup of the general linear group $ G = \mathop{\rm GL} ( n, K) $ has the form

$$ \alpha ( t) = \mathop{\rm exp} tX = \ \sum _ {n = 0 } ^ \infty { \frac{1}{n! } } t ^ {n} X ^ {n} . $$

If $ G $ is a real Lie group endowed with a two-sidedly invariant pseudo-Riemannian metric or affine connection, then the one-parameter subgroups of $ G $ are the geodesics passing through the identity $ e $.

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)
How to Cite This Entry:
One-parameter subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-parameter_subgroup&oldid=11235
This article was adapted from an original article by r group','../u/u095350.htm','Unitary transformation','../u/u095590.htm','Vector bundle, analytic','../v/v096400.htm')" style="background-color:yellow;">A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article