Topological groups, Lie groups Definition

p-adic Lie group

Builds on Topological group p-adic number Power series

Definition

Let $p$ be a prime and let $\Q_p$ be the field of p-adic numbers. Let $U \subseteq \Q_p^n$ be open. A map $f \colon U \to \Q_p^m$ is analytic if each point of $U$ has an open neighbourhood on which each coordinate of $f$ is a convergent power series. A $p$ -adic analytic manifold of dimension $n$ is a Hausdorff space $M$ with an atlas of charts. The charts are homeomorphisms from open sets of $M$ onto open sets of $\Q_p^n$ , and their transition maps are analytic. A $p$ -adic Lie group is a group $G$ with the structure of a $p$ -adic analytic manifold such that:

the multiplication $G \times G \to G$ , $(x, y) \mapsto xy$ , is analytic;
the inversion $G \to G$ , $x \mapsto x^{-1}$ , is analytic.

Remarks

Example: the group $\Z_p$ of $p$ -adic integers under addition is an open subset of $\Q_p$ , so the identity map is a chart. Addition $(x, y) \mapsto x + y$ and negation $x \mapsto -x$ are polynomials, hence analytic. So $\Z_p$ is a $p$ -adic Lie group of dimension $1$ . The group $GL_n(\Q_p)$ is open in the space $\Q_p^{n^2}$ of matrices. Its product and inverse are rational in the entries with nonvanishing denominators, so it is a $p$ -adic Lie group of dimension $n^2$ .

Non-example: the additive group $\R$ is not a $p$ -adic Lie group with its usual topology. Every open subset of $\Q_p^n$ is totally disconnected, so every $p$ -adic analytic manifold is totally disconnected. But $\R$ is connected and has more than one point.

Such groups are also called $p$ -adic analytic groups. Every $p$ -adic Lie group is a totally disconnected locally compact Hausdorff group.

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Power series concept
Topological group concept
p-adic number concept
p-adic Lie group definition

Tag	`CTBNKE0B03`
URI	`m2://CTBNKE0B03`
Kind	definition
Writing	W0 stub
MSC	22E20, 22E35, 20E18
Defines	p-adic Lie group
Uses	Topological group, p-adic number, Power series

p-adic Lie group

Definition

Remarks

Reading path

Further reading

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