---
tag: 3AKM6EPV2F
kind: concept
name: p-adic Lie group
msc: ["22E20", "22E35", "20E18"]
specializes: [92700QAB5Y]
sources:
  - url: https://en.wikipedia.org/wiki/Pro-p_group
---

## Summary

A $p$-adic Lie group is a [[topological group|92700QAB5Y]] that is also an analytic manifold over the field $\Q_p$ of [[p-adic numbers|BT6T702C28]], with analytic group operations.
It is the $p$-adic analogue of a real [[Lie group|7JBRGCVJRB]].
Such groups are totally disconnected, so their structure is studied through compact open subgroups, which are [[pro-p groups|C13D6W3MM9]] up to finite index.
They appear as Galois groups in number theory and as the groups $G(\Q_p)$ of points of algebraic groups.

## Explanations

### L1 human

A real Lie group looks near each point like a piece of $\R^n$, and its multiplication is given by convergent power series.
A $p$-adic Lie group is the same idea with $\R$ replaced by the $p$-adic numbers $\Q_p$.
The additive group $\Z_p$ of $p$-adic integers is the simplest example: it is an open piece of $\Q_p$, and addition is a polynomial.
The matrix group $GL_n(\Q_p)$ is another example, since products and inverses of matrices are given by rational functions of the entries.
Unlike real Lie groups, these groups fall apart into tiny open and closed pieces, because $\Q_p$ itself is totally disconnected.