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I thought they were the same, just different names. Let me make question more precise:

Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}_p$, is $G$ a p-adic Lie group w.r.t. the analytic topology from $\mathbb{Q}_p$ in the sense of Peter Schneider? If this is the case, Does the Lie algebra from the algebraic group coincide with the Lie algebra from the Lie group?

As far as I can see this is true for real number case. But I'm not familiar with p-adic Lie group theory.

p-Adic Lie Groups: Peter Schneider: http://books.google.de/books?id=bjWU3GF93YQC&printsec=frontcover&dq=p-adic%20lie%20groups&hl=de&sa=X&ei=Ml83UcOILpS-9gSLnICYDA&ved=0CDQQ6AEwAA#v=onepage&q=p-adic%20lie%20groups&f=false

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    $\begingroup$ Even over $\mathbf{R}$ this fails: the natural map ${\rm{SL}}_n \rightarrow {\rm{PGL}}_n$ of linear algebraic groups over $\mathbf{R}$ (or any field) is a degree-$n$ isogeny, so not an isomorphism, but on $\mathbf{R}$-points it is an isomorphism of Lie groups. The formation of Lie algebra naturally commutes with analytification of smooth group schemes over any field $F$ complete for a nontrivial absolute value (likewise for the tangent space at an $F$-point on a smooth $F$-scheme). Smooth $F$-groups and analytification in the sense of $F$-points are related but no in sense "the same". $\endgroup$
    – user74230
    Commented Jan 30, 2015 at 3:26
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    $\begingroup$ It would immensely clarify your question if you say what you think the definition of "linear algebraic group over a field $k$" means, and mention your background in algebraic geometry (over non-algebraically closed fields, and with schemes). Also, the notion of a $p$-adic analytic group manifold much predates Peter Schneider; what reference do you have in mind? $\endgroup$
    – user74230
    Commented Jan 30, 2015 at 7:13
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    $\begingroup$ It's true that for every $p$-adic algebraic group $\mathbb{G}$, the group of points $G=\mathbb{G}(\mathbf{Q}_p)$ is a $p$-adic Lie group. See Bourbaki. But plenty of $p$-adic Lie groups don't arise this way, e.g. all discrete groups are $p$-adic Lie groups, and there are many other examples. $\endgroup$
    – YCor
    Commented Jan 30, 2015 at 18:09
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    $\begingroup$ On the other hand when we have a group $G$ as above, we have to be careful saying that $G$ is algebraic because there are several non-equivalent ways... for instance the Zariski topology is not uniquely determined by the structure of topological group. For instance if we consider $G=(\mathbf{Q}_p^*)^2$, it is isomorphic to $(\mathbf{Z}_p^*\times\mathbf{Z})^2$, and thus if we think of its "natural" Zariski topology, there are automorphisms of topological groups (fixing both $\mathbf{Z}_p^*$ and switching both $\mathbf{Z}$), that are not continuous for the Zariski topology (...) $\endgroup$
    – YCor
    Commented Jan 30, 2015 at 18:13
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    $\begingroup$ (...) and the subgroup $\mathbf{Q}_p^*\times\{1\}$ is closed in the Zariski topology while its image is not. Thus for a $p$-adic Lie group that can be "made" algebraic (i.e. isomorphic to some $\mathbb{G}(\mathbf{Q}_p)$), the notion of algebraic subgroup is not intrinsic to ambient topological group. (For this reason Venkataramana's answer makes little sense to me, since $\mathbf{Z}_p$ is not algebraic in a canonical way.) $\endgroup$
    – YCor
    Commented Jan 30, 2015 at 18:17

3 Answers 3

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Consider the map $x\mapsto (x,e^x)$ from $p^2{\mathbb Z}_p$ into ${\mathbb Z}_p\times {\mathbb Z}_p^*$, the latter being the ${\mathbb Z}_p$ rational points of the algebraic group ${\mathbb G}_a\times {\mathbb G}_m$. The image of this map is Zariski dense and hence $p^2{\mathbb Z}_p$ is not an algebraic subgroup of the $p$-adic algebraic group ${\mathbb Z}_p\times {\mathbb Z}_p^*$.

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  • $\begingroup$ I don't really understand your answer. Can you explain more? $\endgroup$
    – m07kl
    Commented Jan 30, 2015 at 11:45
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    $\begingroup$ The argument above shows that $p^2{\mathbb Z}_p$ is a $p$-adic analytic subgroup of ${\mathbb Z}_p\times {\mathbb Z}_p^*$ but is not an algebraic subgroup $\endgroup$
    – Venkataramana
    Commented Jan 30, 2015 at 11:52
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    $\begingroup$ But I ask the opposite direction. Is algebraic group over p-adic number a p-adic Lie group? $\endgroup$
    – m07kl
    Commented Jan 30, 2015 at 11:59
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    $\begingroup$ That is certainly true: algebraic $\implies$ analytic. There are lecture notes by Serre on $p$-adic groups which deal with these things. I don't have the reference at hand, but it is available on Amazon. $\endgroup$
    – Venkataramana
    Commented Jan 30, 2015 at 12:02
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The development of both Lie groups and linear algebraic groups is rather complicated, starting with the definitions over various fields. For example, working over $\mathbb{Q}_p$ or a finite extension is one classical setting, and many of the ideas (though not all) carry over well to "local fields" in prime characteristic.

While ACL and Venkatarama have correctly pointed to Serre's lectures for a conventional treatment of Lie groups over local fields (including $\mathbb{R}$ and "ultrametric" fields), it may help to look more broadly at the development of the ideas over time. Chevalley set out to write a six volume series of books on Lie groups (and "linear algebraic groups"), but abandoned that after three books in order to develop an improved theory of linear algebraic groups using a recent version of algebraic geometry. But he had already realized that compact (real) Lie groups carry a natural algebraic group structure (which yields the same Lie algebra), even though most other Lie groups do not.

His student Robert Hooke at Princeton wrote a thesis in 1942, soon published in the Annals here (available through JSTOR). In this work the standard dictionary between Lie groups and Lie algebras is adapted to $p$-adic groups.

In the next decades Bourbaki (no doubt actively influenced by Serre) began to issue chapters of their treatise Groupes et algebres de Lie, later published in English translation by Springer. Chapter II (Hermann, 1972) lays the foundations for a unified theory of Lie groups and their Lie algebras over $\mathbb{R}, \mathbb{C}$, and complete local fields; sometimes the field is required to be of characteristic 0, so one has to be aware of this. It's very useful to consult their historical notes on Chapters I-III.

Already Serre had lectured at Harvard in 1964 on Lie algebras and Lie groups, borrowing some of the not yet published Bourbaki approach. These lectures were published by W.A. Benjamin (1965) and later reissued as Springer Lecture Notes 1500 here.

Though the theory of $p$-adic Lie groups is now well grounded and to some extent unified with traditional Lie group theory, it remains at some distance from linear algebraic groups. However, as in the real and complex Lie group cases, there are much closer connections when the groups are connected and reductive: this has led to a rich literature, including the work of Iwahori-Matsumoto and Bruhat-Tits on structure theory over local fields. By the time Conrad-Gabber-Prasad wrote their recent book Pseudo-reductive Groups, the group scheme approach allowed further refinements in the study of reductive groups, especially in prime characteristic. (But these developments move farther away from the traditional study of manifolds and Lie groups, including the close relationship between Lie groups and their Lie algebras.)

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As explained in Venkataramana's comment, algebraic groups over $p$-adic fields give rise to $p$-adic Lie groups, and the Lie algebra of the associated $p$-adic Lie groups is the Lie algebra of the initial algebraic group. What makes these things work is a notion of $p$-adic manifolds, which is analogue, but simpler, to the classical notion of analytic real (or complex) manifolds. (It is simpler, because every $p$-adic manifold is a disjoint union of $p$-adic unit balls, due to the fact that its topology is totally discontinuous.)

The standard reference for this fact is probably Serre's book, Lie algebras and Lie groups, Springer Lecture Notes in Mathematics, vol. 1500 (1992).

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