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 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 $( x , y ) \rightarrow x y ^ { - 1 }$ 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) 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 $k$ 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, [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 $D$-adic analytic groups are called, respectively, complex and $D$-adic Lie groups.

The Cartan theorems formulated above signify that the 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 [5]–[9]; if it is $D$-adic, it must contain a finitely generated open subgroup $r$ which is a pro-$D$-group and whose commutator subgroup is contained in the set $U ^ { p ^ { 2 } }$ of $p ^ { 2 }$-th powers of elements in $r$ [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 $D$-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