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A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a0122901.png" /> which possesses at the same time the structure of a [[Topological group|topological group]] and that of a finite-dimensional [[Analytic manifold|analytic manifold]] (over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a0122902.png" /> that is complete in some non-trivial norm, cf. [[Norm on a field|Norm on a field]]) so that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a0122903.png" /> defined by the rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a0122904.png" /> is analytic. An analytic group is always Hausdorff; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a0122905.png" /> is locally compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a0122906.png" /> is locally compact. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a0122907.png" /> is, respectively, the field of real, complex or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a0122908.png" />-adic numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a0122909.png" /> is called a real, complex or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229010.png" />-adic analytic group, respectively. An example of an analytic group is the general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229011.png" /> of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229012.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229013.png" /> (cf. [[Linear classical group|Linear classical group]]) or, more generally, the group of invertible elements of an arbitrary finite-dimensional associative algebra with a unit over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229014.png" />. In general, the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229015.png" />-rational points of an [[Algebraic group|algebraic group]], defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229016.png" />, is an analytic group. A subgroup of an analytic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229017.png" /> which is a submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229018.png" /> is called an analytic subgroup; such a subgroup must be closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229019.png" />. For example, the orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229020.png" /> is an analytic subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229021.png" />. All closed subgroups of a real or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229022.png" />-adic analytic group are analytic, and each continuous homomorphism of such groups is analytic (Cartan's theorems, [[#References|[1]]]).
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A set $  G $  which possesses at the same time the structure of a [[Topological group|topological group]] and that of a finite-dimensional [[Analytic manifold|analytic manifold]] (over a field $  k $  that is complete in some non-trivial norm, cf. [[Norm on a field|Norm on a field]]) so that the mapping $  G \times G \rightarrow G $  defined by the rule $  (x,\  y) \rightarrow xy ^{-1} $  is analytic. An analytic group is always Hausdorff; if $  k $  is locally compact, then $  G $  is locally compact. If $  k $  is, respectively, the field of real, complex or $  p $ -adic numbers, then $  G $  is called a real, complex or $  p $ -adic analytic group, respectively. An example of an analytic group is the general linear group $  \mathop{\rm GL}\nolimits (n,\  k) $  of the vector space $  k ^{n} $  over $  k $  (cf. [[Linear classical group|Linear classical group]]) or, more generally, the group of invertible elements of an arbitrary finite-dimensional associative algebra with a unit over $  k $ . In general, the group of $  k $ -rational points of an [[Algebraic group|algebraic group]], defined over $  k $ , is an analytic group. A subgroup of an analytic group $  G $  which is a submanifold in $  G $  is called an analytic subgroup; such a subgroup must be closed in $  G $ . For example, the orthogonal group $  \textrm{ O }(n,\  k) = \{ {g \in  \mathop{\rm GL}\nolimits (n,\  k)} : {^tgg = 1} \} $  is an analytic subgroup in $  \mathop{\rm GL}\nolimits (n,\  k) $ . All closed subgroups of a real or $  p $ -adic analytic group are analytic, and each continuous homomorphism of such groups is analytic (Cartan's theorems, [[#References|[1]]]).
  
An analytic group is sometimes referred to as a Lie group [[#References|[1]]], but a Lie group is usually understood in the narrower sense of a real analytic group [[#References|[2]]], [[#References|[3]]] (cf. [[Lie group|Lie group]]). Complex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229023.png" />-adic analytic groups are called, respectively, complex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229024.png" />-adic Lie groups.
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An analytic group is sometimes referred to as a Lie group [[#References|[1]]], but a Lie group is usually understood in the narrower sense of a real analytic group [[#References|[2]]], [[#References|[3]]] (cf. [[Lie group|Lie group]]). Complex and $  p $ -adic analytic groups are called, respectively, complex and $  p $ -adic Lie groups.
  
The Cartan theorems formulated above signify that the [[Category|category]] of real or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229025.png" />-adic analytic groups is a complete subcategory in the category of locally compact topological groups. The question of the extent to which these categories differ, i.e. as to when a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229026.png" /> is a real analytic or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229027.png" />-adic analytic group, can be exhaustively answered: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229028.png" /> is real analytic, it must contain a neighbourhood of the unit without non-trivial subgroups [[#References|[5]]]–[[#References|[9]]]; if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229029.png" />-adic, it must contain a finitely generated open subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229030.png" /> which is a [[Pro-p group|pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229031.png" />-group]] and whose commutator subgroup is contained in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229033.png" />-th powers of elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229034.png" /> [[#References|[10]]]. In particular, any topological group with a neighbourhood of the unit that is homeomorphic to a Euclidean space (a so-called locally Euclidean topological group, [[#References|[4]]]) is a real analytic group. In other words, if continuous local coordinates exist in a topological group, it follows that analytic local coordinates exist; this result is the positive solution of Hilbert's fifth problem [[#References|[5]]], [[#References|[11]]].
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The Cartan theorems formulated above signify that the [[Category|category]] of real or $  p $ -adic analytic groups is a complete subcategory in the category of locally compact topological groups. The question of the extent to which these categories differ, i.e. as to when a locally compact group $  G $  is a real analytic or a $  p $ -adic analytic group, can be exhaustively answered: If $  G $  is real analytic, it must contain a neighbourhood of the unit without non-trivial subgroups [[#References|[5]]]–[[#References|[9]]]; if it is $  p $ -adic, it must contain a finitely generated open subgroup $  U $  which is a [[Pro-p group|pro- $  p $ -group]] and whose commutator subgroup is contained in the set $  U ^ {p ^{2}} $  of $  p ^{2} $ -th powers of elements in $  U $  [[#References|[10]]]. In particular, any topological group with a neighbourhood of the unit that is homeomorphic to a Euclidean space (a so-called locally Euclidean topological group, [[#References|[4]]]) is a real analytic group. In other words, if continuous local coordinates exist in a topological group, it follows that analytic local coordinates exist; this result is the positive solution of Hilbert's fifth problem [[#References|[5]]], [[#References|[11]]].
  
If the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229035.png" /> is zero, the most important method in the study of analytic groups is the study of their Lie algebras (cf. [[Lie algebra of an analytic group|Lie algebra of an analytic group]]).
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If the characteristic of the field $  k $  is zero, the most important method in the study of analytic groups is the study of their Lie algebras (cf. [[Lie algebra of an analytic group|Lie algebra of an analytic group]]).
  
 
For infinite-dimensional analytic groups cf. [[Lie group, Banach|Lie group, Banach]].
 
For infinite-dimensional analytic groups cf. [[Lie group, Banach|Lie group, Banach]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)  {{MR|0218496}} {{ZBL|0132.27803}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)    {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Chevalley,  "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)  {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Helgason,  "Differential geometry and symmetric spaces" , Acad. Press  (1962)  {{MR|0145455}} {{ZBL|0111.18101}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  "Hilbert problems"  ''Bull. Amer. Math. Soc.'' , '''8'''  (1902)  pp. 101–115  (Translated from German)   {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.M. Gleason,  "Groups without small subgroups"  ''Ann. of Math. (2)'' , '''56''' :  2  (1952)  pp. 193–212  {{MR|0049203}} {{ZBL|0049.30105}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D. Montgomery,  L. Zippin,  "Small subgroups for finite dimensional groups"  ''Ann. of Math. (2)'' , '''56''' :  2  (1952)  pp. 213–241 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  H. Yamabe,  "On the conjecture of Iwasawa and Gleason"  ''Ann. of Math. (2)'' , '''58''' :  1  (1953)  pp. 48–54  {{MR|0054613}} {{ZBL|0053.01601}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Yamabe,  "A generalization of a theorem of Gleason"  ''Ann. of Math. (2)'' , '''58''' :  2  (1953)  pp. 351–365  {{MR|0058607}} {{ZBL|0053.01602}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M. Lazard,  "Groupes analytiques <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012290/a01229036.png" />-adiques"  ''Publ. Math. IHES'' , '''26'''  (1965)  {{MR|209286}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  I. Kaplansky,  "Lie algebras and locally compact groups" , Chicago Univ. Press  (1971)  {{MR|0276398}} {{ZBL|0223.17001}} </TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)  {{MR|0218496}} {{ZBL|0132.27803}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)    {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Chevalley,  "Theory of Lie groups" , '''1''' , Princeton Univ. Press  (1946)  {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Helgason,  "Differential geometry and symmetric spaces" , Acad. Press  (1962)  {{MR|0145455}} {{ZBL|0111.18101}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  "Hilbert problems"  ''Bull. Amer. Math. Soc.'' , '''8'''  (1902)  pp. 101–115  (Translated from German)       {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.M. Gleason,  "Groups without small subgroups"  ''Ann. of Math. (2)'' , '''56''' :  2  (1952)  pp. 193–212  {{MR|0049203}} {{ZBL|0049.30105}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D. Montgomery,  L. Zippin,  "Small subgroups for finite dimensional groups"  ''Ann. of Math. (2)'' , '''56''' :  2  (1952)  pp. 213–241     {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  H. Yamabe,  "On the conjecture of Iwasawa and Gleason"  ''Ann. of Math. (2)'' , '''58''' :  1  (1953)  pp. 48–54  {{MR|0054613}} {{ZBL|0053.01601}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Yamabe,  "A generalization of a theorem of Gleason"  ''Ann. of Math. (2)'' , '''58''' :  2  (1953)  pp. 351–365  {{MR|0058607}} {{ZBL|0053.01602}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M. Lazard,  "Groupes analytiques a01229036.png-adiques"  ''Publ. Math. IHES'' , '''26'''  (1965)  {{MR|209286}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  I. Kaplansky,  "Lie algebras and locally compact groups" , Chicago Univ. Press  (1971)  {{MR|0276398}} {{ZBL|0223.17001}} </TD></TR></table>
  
  

Latest revision as of 17:45, 12 December 2019

A set $ G $ which possesses at the same time the structure of a topological group and that of a finite-dimensional analytic manifold (over a field $ k $ that is complete in some non-trivial norm, cf. Norm on a field) so that the mapping $ G \times G \rightarrow G $ defined by the rule $ (x,\ y) \rightarrow xy ^{-1} $ is analytic. An analytic group is always Hausdorff; if $ k $ is locally compact, then $ G $ is locally compact. If $ k $ is, respectively, the field of real, complex or $ p $ -adic numbers, then $ G $ is called a real, complex or $ p $ -adic analytic group, respectively. An example of an analytic group is the general linear group $ \mathop{\rm GL}\nolimits (n,\ k) $ of the vector space $ k ^{n} $ over $ k $ (cf. Linear classical group) or, more generally, the group of invertible elements of an arbitrary finite-dimensional associative algebra with a unit over $ k $ . In general, the group of $ k $ -rational points of an algebraic group, defined over $ k $ , is an analytic group. A subgroup of an analytic group $ G $ which is a submanifold in $ G $ is called an analytic subgroup; such a subgroup must be closed in $ G $ . For example, the orthogonal group $ \textrm{ O }(n,\ k) = \{ {g \in \mathop{\rm GL}\nolimits (n,\ k)} : {^tgg = 1} \} $ is an analytic subgroup in $ \mathop{\rm GL}\nolimits (n,\ k) $ . All closed subgroups of a real or $ p $ -adic analytic group are analytic, and each continuous homomorphism of such groups is analytic (Cartan's theorems, [1]).

An analytic group is sometimes referred to as a Lie group [1], but a Lie group is usually understood in the narrower sense of a real analytic group [2], [3] (cf. Lie group). Complex and $ p $ -adic analytic groups are called, respectively, complex and $ p $ -adic Lie groups.

The Cartan theorems formulated above signify that the category of real or $ p $ -adic analytic groups is a complete subcategory in the category of locally compact topological groups. The question of the extent to which these categories differ, i.e. as to when a locally compact group $ G $ is a real analytic or a $ p $ -adic analytic group, can be exhaustively answered: If $ G $ is real analytic, it must contain a neighbourhood of the unit without non-trivial subgroups [5][9]; if it is $ p $ -adic, it must contain a finitely generated open subgroup $ U $ which is a pro- $ p $ -group and whose commutator subgroup is contained in the set $ U ^ {p ^{2}} $ of $ p ^{2} $ -th powers of elements in $ U $ [10]. In particular, any topological group with a neighbourhood of the unit that is homeomorphic to a Euclidean space (a so-called locally Euclidean topological group, [4]) is a real analytic group. In other words, if continuous local coordinates exist in a topological group, it follows that analytic local coordinates exist; this result is the positive solution of Hilbert's fifth problem [5], [11].

If the characteristic of the field $ k $ is zero, the most important method in the study of analytic groups is the study of their Lie algebras (cf. Lie algebra of an analytic group).

For infinite-dimensional analytic groups cf. Lie group, Banach.

References

[1] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[2] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[3] C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) MR0082628 MR0015396 Zbl 0063.00842
[4] S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101
[5] "Hilbert problems" Bull. Amer. Math. Soc. , 8 (1902) pp. 101–115 (Translated from German)
[6] A.M. Gleason, "Groups without small subgroups" Ann. of Math. (2) , 56 : 2 (1952) pp. 193–212 MR0049203 Zbl 0049.30105
[7] D. Montgomery, L. Zippin, "Small subgroups for finite dimensional groups" Ann. of Math. (2) , 56 : 2 (1952) pp. 213–241
[8] H. Yamabe, "On the conjecture of Iwasawa and Gleason" Ann. of Math. (2) , 58 : 1 (1953) pp. 48–54 MR0054613 Zbl 0053.01601
[9] H. Yamabe, "A generalization of a theorem of Gleason" Ann. of Math. (2) , 58 : 2 (1953) pp. 351–365 MR0058607 Zbl 0053.01602
[10] M. Lazard, "Groupes analytiques a01229036.png-adiques" Publ. Math. IHES , 26 (1965) MR209286
[11] I. Kaplansky, "Lie algebras and locally compact groups" , Chicago Univ. Press (1971) MR0276398 Zbl 0223.17001


Comments

In Western literature a connected Lie group is often called an analytic group.

Cartan's theorems usually go back to J. von Neumann (cf. [a1], [a2]).

References

[a1] J. von Neumann, , Collected works , 1 , Pergamon (1961) pp. 134–148 Zbl 0188.00102 Zbl 0100.00202
[a2] J. von Neumann, , Collected works , 1 , Pergamon (1961) pp. 509–548 Zbl 0188.00102 Zbl 0100.00202
How to Cite This Entry:
Analytic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_group&oldid=21805
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article