Application of simple Lie algebras over finite fields I am now interested in simple Lie algebras over finite fields. In Lie algebras over the complex numbers, there are several applications and some related topics.
Is there any potential application for simple Lie algebras over finite fields, or anything related? Perhaps, in coding theory or graph theory?
 A: Fourier transform of invariant functions on finite reductive Lie algebras have been developed in here: http://www.springer.com/de/book/9783540240204
This can be considered the Lie algebra analogue of character tables of finite groups
The MacWilliams identities in error correcting codes https://www.encyclopediaofmath.org/index.php/MacWilliams_identities
have a Fourier transform interpretation for example here:
https://www.jstor.org/stable/pdf/25098937.pdf 
This can considered Fourier transform on abelian Lie algebras.
Finally Fourier transform on finite Lie algebras was used in the proof of a conjecture of Kac in the representation theory of quivers
https://arxiv.org/abs/1204.2375
A: I apologize for the self-promotion, but this is by the request of the OP. Below are several references. I should say that they are applications for pro-$p$ groups. So I cannot think about examples for $p$-groups now. Also, I am a bit liberal in the sense that most of them use loop algebras that are built from simple Lie algebras over finite fields. Finally, I cannot remember how much Lie algebra is used on the last one.


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*Lubotzky, Alexander; Shalev, Aner On some Λ-analytic pro-p groups.

*Barnea, Yiftach; Shalev, Aner Hausdorff dimension, pro-p groups, and Kac-Moody algebras.

*Barnea, Yiftach Residual properties of free pro-p groups.

*Barnea, Yiftach; Guralnick, Robert Subgroup growth in some pro-p groups.

*Barnea, Yiftach Generators of simple Lie algebras and the lower rank of some pro-p-groups.

*Barnea, Yiftach; Klopsch, Benjamin Index-subgroups of the Nottingham group.

*Ershov, Mikhail On the commensurator of the Nottingham group.

*Ershov, Mikhail Finite presentability of SL1(D).

*Ershov, Mikhail On subgroups of the Nottingham group of positive Hausdorff dimension.

*Ershov, M. V. The Nottingham group is finitely presented.

*Ershov, Mikhail New just-infinite pro-p groups of finite width and subgroups of the Nottingham group.

*Abért, Miklós; Nikolov, Nikolay; Szegedy, Balázs Congruence subgroup growth of arithmetic groups in positive characteristic.

A: I can add one more example from geometry, topology and physics, where simple modular Lie algebras arise. In John Milnor's article On Fundamental Groups of Complete Affinely Flat Manifolds there was an open question for Lie algebras $L$, namely which Lie algebras admit a bijective $1$-cocycle in $Z^1(L,M)$ with $\dim(L)=\dim (M)$. For the adjoint module, this asks for a non-singular derivation. Jacobson proved that in characteristic zero such a Lie algebra must be nilpotent.
The question was formulated there also for Lie groups, namely which connected Lie groups admit a left-invariant affine structure. It can also be formulated in terms of Yang-Baxter groups, or left braces, see the article Counterexample to a conjecture about braces by D. Bachiller. 
From the first Whitehead Lemma it follows that a simple Lie algebra of characteristic zero does not admit such a $1$-cocycle. In prime characteristic, this is no longer true. For modular simple Lie algebras there are such examples, see First Cohomology Groups for Classical Lie Algebras by Jens-Carsten Jantzen.
 This has then interesting applications to $p$-groups, Yang-Baxter groups, left braces, pre-Lie algebras, post-Lie algebras, and other things.
Benkart, Kostrikin and Kuznezov classified Finite-Dimensional Modular Simple Lie algebras with a Nonsingular Derivation.
