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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.
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{{TEX|done}}{{MSC|22E20}}
 
 
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>
 
  
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An [[analytic group]] over the field $\mathbb{Q}_p$ of [[p-adic number|$p$-adic number]]s (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 number]]s 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.
  
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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====
 
====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]]].
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For the representation theory of reductive $p$-adic groups see [[#References|[a1]]], [[#References|[a2]]].
  
 
====References====
 
====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>
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<table>
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<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>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  M. Lazard,  "Groupes analytiques $p$-adiques"  ''Publ. Math. IHES'' , '''26'''  (1965)  pp. 389–603</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  Harish-Chandra,  "Collected papers" , '''1–4''' , Springer  (1984)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.J. Silberger,  "Introduction to harmonic analysis on reductive $p$-adic groups" , Princeton Univ. Press  (1979)</TD></TR>
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<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>
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</table>

Latest revision as of 11:17, 9 April 2023

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, p-adic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_p-adic&oldid=18653
This article was adapted from an original article by A.A. Kirillov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article