View source for Lie group, p-adic

An [[Analytic group|analytic group]] over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l0586702.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l0586703.png" />-adic numbers (more generally, over a locally compact non-Archimedean field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l0586704.png" />). Natural examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l0586705.png" />-adic Lie groups are the Galois groups of certain infinite extensions of fields. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l0586706.png" /> is the field obtained by adjoining to the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l0586707.png" /> of rational numbers a primitive root of unity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l0586708.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l0586709.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867011.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867012.png" /> the Galois group of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867013.png" /> is isomorphic to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867014.png" />-adic Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867015.png" />, the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867016.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867017.png" />-adic case. These results have been applied in algebraic number theory and in group theory.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,   "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Lazard,   "Groupes analytiques <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867018.png" />-adiques"  ''Publ. Math. IHES'' , '''26'''  (1965)  pp. 389–603</TD></TR></table>



====Comments====
For the representation theory of reductive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867019.png" />-adic groups see [[#References|[a1]]], [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Harish-Chandra,   "Collected papers" , '''1–4''' , Springer  (1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.J. Silberger,   "Introduction to harmonic analysis on reductive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058670/l05867020.png" />-adic groups" , Princeton Univ. Press  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Burhat,   J. Tits,   "Groupes réductifs sur un corps local"  ''Publ. Math. IHES'' , '''41'''  (1972)  pp. 5–251</TD></TR></table>