Analytic group

A set which possesses at the same time the structure of a topological group and that of a finite-dimensional analytic manifold (over a field that is complete in some non-trivial norm, cf. Norm on a field) so that the mapping defined by the rule is analytic. An analytic group is always Hausdorff; if is locally compact, then is locally compact. If is, respectively, the field of real, complex or -adic numbers, then is called a real, complex or -adic analytic group, respectively. An example of an analytic group is the general linear group of the vector space over (cf. Linear classical group) or, more generally, the group of invertible elements of an arbitrary finite-dimensional associative algebra with a unit over . In general, the group of -rational points of an algebraic group, defined over , is an analytic group. A subgroup of an analytic group which is a submanifold in is called an analytic subgroup; such a subgroup must be closed in . For example, the orthogonal group is an analytic subgroup in . All closed subgroups of a real or -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 -adic analytic groups are called, respectively, complex and -adic Lie groups.

The Cartan theorems formulated above signify that the category of real or -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 is a real analytic or a -adic analytic group, can be exhaustively answered: If is real analytic, it must contain a neighbourhood of the unit without non-trivial subgroups [5]–[9]; if it is -adic, it must contain a finitely generated open subgroup which is a pro--group and whose commutator subgroup is contained in the set of -th powers of elements in [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 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)
[2]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[3]	C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946)
[4]	S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962)
[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
[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
[9]	H. Yamabe, "A generalization of a theorem of Gleason" Ann. of Math. (2) , 58 : 2 (1953) pp. 351–365
[10]	M. Lazard, "Groupes analytiques -adiques" Publ. Math. IHES , 26 (1965)
[11]	I. Kaplansky, "Lie algebras and locally compact groups" , Chicago Univ. Press (1971)

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
[a2]	J. von Neumann, , Collected works , 1 , Pergamon (1961) pp. 509–548