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''of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682001.png" /> over a normed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682002.png" />''
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An analytic homomorphism of the additive group of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682003.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682004.png" />, that is, an analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682005.png" /> such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682006.png" /></td> </tr></table>
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''of a Lie group  $  G $
 +
over a normed field  $  K $''
  
The image of this homomorphism, which is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682007.png" />, is also called a one-parameter subgroup. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682008.png" />, then the continuity of the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o0682009.png" /> implies that it is analytic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820011.png" />, then for any tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820012.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820013.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820014.png" /> there exists a unique one-parameter subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820015.png" /> having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820016.png" /> as its tangent vector at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820017.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820020.png" /> is the [[Exponential mapping|exponential mapping]]. In particular, any one-parameter subgroup of the [[General linear group|general linear group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820021.png" /> has the form
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An analytic homomorphism of the additive group of the field  $  K $
 +
into  $  G $,
 +
that is, an analytic mapping $  \alpha : K \rightarrow G $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820022.png" /></td> </tr></table>
+
$$
 +
\alpha ( s + t)  = \alpha ( s) \alpha ( t),\  s, t \in K.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820023.png" /> is a real Lie group endowed with a two-sidedly invariant pseudo-Riemannian metric or affine connection, then the one-parameter subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820024.png" /> are the geodesics passing through the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068200/o06820025.png" />.
+
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|exponential mapping]]. In particular, any one-parameter subgroup of the [[General linear group|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Helgason,  "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , Hermann  (1972)  pp. Chapt. 2; 3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Hochschild,  "Structure of Lie groups" , Holden-Day  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , Hermann  (1972)  pp. Chapt. 2; 3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Hochschild,  "Structure of Lie groups" , Holden-Day  (1965)</TD></TR></table>

Latest revision as of 08:04, 6 June 2020


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=48042
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