# 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

 [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