# 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, ).

An analytic group is sometimes referred to as a Lie group , but a Lie group is usually understood in the narrower sense of a real analytic group ,  (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 ; 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 $U$ . 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, ) 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 , .

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.

How to Cite This Entry:
Analytic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_group&oldid=44227
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article