# Lie group, p-adic

An analytic group over the field of -adic numbers (more generally, over a locally compact non-Archimedean field ). Natural examples of -adic Lie groups are the Galois groups of certain infinite extensions of fields. For example, if is the field obtained by adjoining to the field of rational numbers a primitive root of unity of degree and , , then for the Galois group of the extension is isomorphic to the -adic Lie group , the group of -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 -adic case. These results have been applied in algebraic number theory and in group theory.

#### 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 -adiques" Publ. Math. IHES , 26 (1965) pp. 389–603 |

#### Comments

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

#### References

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

[a2] | A.J. Silberger, "Introduction to harmonic analysis on reductive -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, p-adic.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_p-adic&oldid=18653