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Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ is triangularizable. Is this result still true, in other words does Lie's theorem remain true, if $K$ has characteristic $p$ where $p>n$ ? Does there exist some reference ?

For instance, the Jacobson's lemma, related to nilpotent matrices, is valid if $p>n$. More generally, what do you think about the following assertion: results related to Lie algebras, that are true in characteristic $0$, are true in characteristic $p$ provided we stick to vector spaces of dimension $n<p$? I ask the question because, very often in linear algebra papers, the underlying field is $\mathbb{C}$; yet, quite often, these results can be generalized to fields of characteristic $0$ and even of characteristic $p$. In particular, the question arises whether an algorithm written for a field with $\mathrm{char}(K)=0$ can also apply when $\mathrm{char}(K)=p>0$. In general, when I ask authors I get no clear answer.

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  • $\begingroup$ This does not answer your question, but at least this holds in characteristic large enough, say $p\ge p_n$. Indeed if it fails for some given $n$, the existence for large $p_i$ of a solvable Lie algebra in dimension $n$ with no invariant line yields a solvable Lie algebra in dimension $n$ with no invariant line on the ultraproduct of the fields. But this provides no explicit bound about $p_n$ and the one you suggest looks plausible. $\endgroup$
    – YCor
    Commented Apr 22, 2014 at 17:30

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Lie’s theorem indeed still holds in positive characteristic provided the dimension of the vector space is less than the characteristic. For reference see, for example, the remark before example $81$ in http://math.berkeley.edu/~reb/courses/261/11.pdf, It is indeed often the case that results being true in characteristic zero, remain true for $p$ larger than the dimension of the vector space (or larger than the Coxeter number for simple Lie algebras). On the other hand there may be many results from the modular world where we need to be careful with such a statement. The classification of simple Lie algebras from characteristic zero does not remain true for large $p$.

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  • $\begingroup$ still the classification of simple Lie algebras in dimension $\le n$ remains true for $p\ge p_n$, for some $p_n$, isn't it? $\endgroup$
    – YCor
    Commented Apr 22, 2014 at 18:32
  • $\begingroup$ Yes, I think this is correct. If $p\ge n$ then a simple modular Lie algebra of dimension $n$ is classical or isomorphic to $W(1;1)$, the Witt algebra of dimension $p$. $\endgroup$
    – Dietrich Burde
    Commented Apr 22, 2014 at 18:43
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    $\begingroup$ OK good; 1st order logic (and the finiteness of the number of simple Lie algebras over $\mathbf{C}$) already shows that the classification in dimension $n$ is the same as in characteristic zero for large characteristic, say for $p\ge p_n$; your reply shows that we can choose $p_n=n+1$ and this is optimal when $n$ is prime. $\endgroup$
    – YCor
    Commented Apr 22, 2014 at 19:19
  • $\begingroup$ Ditrich, thanks for the comment. I note that Lie's and Jacobson's theorems are valid when $char(K)>n$. I find it curious that this is rarely explicitly stated in most of the references; do there exist other theorems concerning lie algebras with this property? $\endgroup$
    – loup blanc
    Commented Apr 24, 2014 at 12:18
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    $\begingroup$ @loupblanc Another interesting theorem is the Jacobson-Morozov theorem. For a discussion in characteristic $p$ see mathoverflow.net/questions/105781/…. Again the counterexample is for a vector space of dimension $n$ and $p=n$. $\endgroup$
    – Dietrich Burde
    Commented Apr 24, 2014 at 12:22
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This is addressed in G. Seligman's Modular Lie Algebras, chapter V §1, with more references therein. In particular Seligman writes "some of the proofs referred to above [among them is Jacobson's in Lie Algebras, pp. 43--50] are still applicable when the degree of the matrices is less than the characteristic". He also discusses (non-)validity of some corollaries of Lie's theorem in prime characteristic.

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  • $\begingroup$ Torsten, thanks for the reference. In fact Seligman is not clear about whether Lie's theorem (or another one) is still valid if the characteristic is greater than $n$. For instance, he gives a counter-example when $char(K)=n$, using the fact that there are $A,B$ s.t. $AB-BA=I$ (that is not a scoop); moreover he gives negative results but, really, no explicit positive results, that is discouraging. $\endgroup$
    – loup blanc
    Commented Apr 24, 2014 at 11:57

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