Over the decades there has been a lot of papers devoted to the classification of Lie algebras of low dimension. Do you know any paper dealing with the problem of determining (up to restricted isomorphisms) restricted Lie algebras $(L,[p])$ of low dimension over a field of characteristic $p>0$?
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asked Aug 6, 2011 at 11:03
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3 Answers
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As far as I know this type of classification problem has attracted very limited attention over the years. That's probably due in part to lack of enough external motivation, coupled with the cautionary example of rapid growth in numbers of nonisomorphic Lie algebras in characteristic 0 as the dimension increases (nilpotent Lie algebras for instance). There are some algorithmic possibilities in prime characteristic nowadays. A treatise by Willem de Graaf covers a lot of ground but may not help directly with your specific question: Lie algebras: theory and algorithms. North-Holland Mathematical Library, 56. North-Holland Publishing Co., Amsterdam, 2000.
The main output of papers on classification in recent decades has involved simple Lie algebras (restricted or not) in prime characteristic, with the most definitive solution found for $p>3$ by Premet and Strade in a series of papers (and monographs).
answered Aug 6, 2011 at 15:48
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$\begingroup$ Dear Professor Humphreys, thank you very much for your answer. Of course, I know the monumental work of Block-Wilson-Strade-Premet on the classification of finite-dimensional simple Lie algebra over algebraically closed field of characteristic $p>3$, and I am aware of the work of W. de Graaf (in particular, about nilpotent Lie algebras up to dimension 6 and solvable Lie algebras up to dimension 4 over arbitrary fields) and other authors. On the other hand, I found very few about the restricted version of this problem, and I perfectly agree with you about the possible reasons why this happens. $\endgroup$ Commented Aug 6, 2011 at 17:21
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Only for information, I would like mentioning that a paper of C. Schneider and H. Usefi just on this topic recently appeared on arXiv: http://arxiv.org/abs/1404.1047
answered Jul 11, 2014 at 14:21
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H. Strade, Lie algebras of small dimension, Contemp. Math. 442 (2007), 232--265; arXiv:math/0601413 - classifies nonsolvable Lie algebras of dimension <7 over a finite field. I think there is a corresponding GAP package.
Another seemingly relevant reference: M. Vaughan-Lee, Simple Lie algebras of low dimension over GF(2), LMS J. Comput. Math. 9 (2006), 174-192.
answered Aug 8, 2011 at 21:12
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$\begingroup$ Dear Pasha, thanks for your answer. I also knew these papers, however they are just concerning with the ordinary case and not with the restricted one. Of course, if you have a Lie algebra with zero center (in particular, a semisimple Lie algebra) you can define at most a p-map on it. But what can one say in general on the isomorphism classes of the restricted Lie algebras of a fixed dimension? $\endgroup$ Commented Aug 9, 2011 at 0:14
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$\begingroup$ Well, I think it would be not difficult to infer this information from the description of (isomorphism classess of) all low-dimensional modular Lie algebras. $\endgroup$ Commented Aug 9, 2011 at 6:23
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$\begingroup$ In general, you can define a lot of $p$-maps on a restrictable Lie algebra $L$ and obtain several classes of isomorphisms of restricted Lie algebras. Indeed, if $[p]$ is a $p$-map on $L$ then $[p]+f$ is also a $p$-map of $L$ for every $p$-semilinear map $f:L⟶Z(L)$, where $Z(L)$ is the center of $L$. (Moreover, every $p$-map of $L$ can be obtained in this way.) Now, one should be able to classify the p-semilinear maps $f$ such that $(L,[p])$ and $(L,[p]+f)$ are isomorphic as restricted Lie algebra, but in general this do not seem to be an easy task. $\endgroup$ Commented Aug 9, 2011 at 10:08