View source for User:Maximilian Janisch/latexlist/Algebraic Groups/Analytic group

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A set $k$ 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 $( x , y ) \rightarrow x y ^ { - 1 }$ is analytic. An analytic group is always Hausdorff; if $k$ is locally compact, then $k$ is locally compact. If $k$ is, respectively, the field of real, complex or $D$-adic numbers, then $k$ is called a real, complex or $D$-adic analytic group, respectively. An example of an analytic group is the general linear group $GL ( n , k )$ 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 $k$ which is a submanifold in $k$ is called an analytic subgroup; such a subgroup must be closed in $k$. For example, the orthogonal group $O ( n , k ) = \{ g \in GL ( n , k ) : \square ^ { t } g g = 1 \}$ is an analytic subgroup in $GL ( n , k )$. All closed subgroups of a real or $D$-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 $D$-adic analytic groups are called, respectively, complex and $D$-adic Lie groups.

The Cartan theorems formulated above signify that the [[Category|category]] of real or $D$-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 $k$ is a real analytic or a $D$-adic analytic group, can be exhaustively answered: If $k$ is real analytic, it must contain a neighbourhood of the unit without non-trivial subgroups [[#References|[5]]]–[[#References|[9]]]; if it is $D$-adic, it must contain a finitely generated open subgroup $r$ which is a [[Pro-p group|pro-$D$-group]] and whose commutator subgroup is contained in the set $U ^ { p ^ { 2 } }$ of $p ^ { 2 }$-th powers of elements in $r$ [[#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 $k$ 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]].

====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 $D$-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>



====Comments====
In Western literature a connected [[Lie group|Lie group]] is often called an analytic group.

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

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  J. von Neumann,  , ''Collected works'' , '''1''' , Pergamon  (1961)  pp. 134–148  {{MR|}} {{ZBL|0188.00102}} {{ZBL|0100.00202}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. von Neumann,  , ''Collected works'' , '''1''' , Pergamon  (1961)  pp. 509–548  {{MR|}} {{ZBL|0188.00102}} {{ZBL|0100.00202}} </td></tr></table>