# Lie group, p-adic

2020 Mathematics Subject Classification: *Primary:* 22E20 [MSN][ZBL]

An analytic group over the field $\mathbb{Q}_p$ of $p$-adic numbers (more generally, over a locally compact non-Archimedean field $K$). Natural examples of $p$-adic Lie groups are the Galois groups of certain infinite extensions of fields. For example, if $\mathbb{Q}(\zeta_{p^\nu})$ is the field obtained by adjoining to the field $\mathbb{Q}$ of rational numbers a primitive root of unity $\zeta_{p^\nu}$ of order $p^\nu$ and $k = \mathbb{Q}(\zeta_p)$, $K = \bigcup_{\nu=1}^\infty \mathbb{Q}(\zeta_{p^\nu})$, then for $p \neq 2$ the Galois group of the extension $K/k$ is isomorphic to the $p$-adic Lie group $\mathbb{Z}_{p}$, the group of $p$-adic integers.

Many results in the theory of ordinary Lie groups (the connection between Lie groups and Lie algebras, the construction and properties of the exponential mapping) have analogues in the $p$-adic case. These results have been applied in algebraic number theory and in group theory.

#### Comments

For the representation theory of reductive $p$-adic groups see [a1], [a2].

#### References

[1] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) |

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

[3] | M. Lazard, "Groupes analytiques $p$-adiques" Publ. Math. IHES , 26 (1965) pp. 389–603 |

[a1] | Harish-Chandra, "Collected papers" , 1–4 , Springer (1984) |

[a2] | A.J. Silberger, "Introduction to harmonic analysis on reductive $p$-adic groups" , Princeton Univ. Press (1979) |

[a3] | F. Burhat, J. Tits, "Groupes réductifs sur un corps local" Publ. Math. IHES , 41 (1972) pp. 5–251 |

**How to Cite This Entry:**

Lie-group-adic.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lie-group-adic&oldid=38704