Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3... - MathOverflow

Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3...

2012-11-17T20:19:12Z

Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem.

I wonder what is known/expected for char p=2,3 ?

More vague and soft question is the following - look at some famous classification problems: simple finite-dim Lie algebras, simple finite groups, some other things classified by ADE... We see the following pattern: there are some series of objects and finite number of "sporadic" objects. I.e. it never happens that there is infinite number of examples which are not in "series". So classification of simple objects is simple (in some very informal sense).

The question: can we expect this in advance, without obtaining classification ? (What are other examples/counter examples of similar phenomenon ?).

For example can we expect/prove this for simple Lie algs for char =2,3 ? I.e. there will be some finite number of series and finite number of "sporadic" examples ?

Answer by Salvatore Siciliano for Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3...

2012-11-21T17:34:10Z

According with the introduction of Strade's book "Simple Lie algebras over fields of positive characteristic. Structure Theory", it seems that a possible list of known finite-dimensional simple Lie algebras over algebraically closed fields of characteristic 3 could be close to complete. A discussion on this topics can be found in Section 4.4 of that book. On the other hand, in characteristic 2 the situation seems to be more complicated. For example, in the paper [Yu. Kochetov - D. Leites: Simple Lie algebras of characteristic 2 recovered from superalgebras and on the notion of a simple group, in Proceedings of the International Algebraic Conference in the Memory of A.I. Malcev, Novosibirsk, Contemp. Math. 131 (1992), 59-67] the authors have constructed simple Lie algebras in characteristic 2 from superalgebras. Thus one expects that a greater variety of constructions could get many more examples in this exceptional characteristic.

Answer by Pasha Zusmanovich for Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3...

2013-05-10T13:32:20Z

(This is too long for a comment). There is a recent surge of activity around (attempts of) classification of simple finite-dimensional Lie algebras in $p=2,3$. It is my understanding that the common view among experts is that the case $p=3$ might be in sight, while situation in $p=2$ is still chaotic. The main current players in the field are A. Grishkov, M.I. Kuznetsov and his students, and D. Leites and his collaborators.

As in Kostrikin-Shafarevich program for large characteristics, deformations (of some initial set of algebras) play a role. Computers are involved a lot.

A sample of recent publications:

S. Bouarroudj, P. Grozman, D. Leites, Infinitesimal deformations of symmetric simple modular Lie algebras and Lie superalgebras, arXiv:0807.3054
S. Bouarroudj, A. Lebedev, D. Leites, I. Shchepochkina, Deforms of Lie algebras in characteristic 2: semi-trivial for Jurman algebras, non-trivial for Kaplansky algebras, arXiv:1301.2781.
B. Eick, Some new simple Lie algebras in characteristic 2, J. Symb. Comput. 45 (2010), N9, 943-951
A. Grishkov, On simple Lie algebras over a field of characteristic 2, J. Algebra 363 (2012), 14-18
A. Grishkov and M. Guerreiro, On simple Lie algebras of dimension seven over fields of characteristic 2, Sao Paulo J. Math. Sci. 4 (2010), N1, 93--107
D. Leites, Towards classification of simple finite dimensional modular Lie superalgebras, arXiv:0710.5638.
M. Vaughan-Lee, Simple Lie algebras of low dimension over GF(2), LMS J. Comput. Math. 9 (2006), 174--192

Answer by Dietrich Burde for Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3...

2013-05-10T15:49:21Z

In general, the classification of finite-dimensional simple non-associative algebras is difficult, i.e., not known, even for algebraically closed fields of characteristic zero. I have tried to start a classification of all complex simple pre-Lie algebras. The first results show that a classification will be very difficult (see http://homepage.univie.ac.at/Dietrich.Burde/papers/burde_08_simple_lsa.pdf). The special case of complex simple Novikov algebras has been solved by E. Zelmanov.