Analytic group

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[edit]

[1] |  J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803

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[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[edit]

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[edit]

[a1] |  J. von Neumann, , Collected works , 1 , Pergamon (1961) pp. 134–148 Zbl 0188.00102 Zbl 0100.00202

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[a2] |  J. von Neumann, , Collected works , 1 , Pergamon (1961) pp. 509–548 Zbl 0188.00102 Zbl 0100.00202