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New Simple Lie Algebras: Melikyan Algebras

It was realized 150 years ago that one of the main problems in the study of Lie algebras would be the identification of the simple ones. Over the field of complex numbers, the simple Lie algebras were found by Killing-Cartan. This nearly 100-year-old classification theorem for simple Lie algebras over algebraically closed fields of characteristic 0 states that the simple Lie algebras of types An, Bn, Cn, Dn, along with 5 exceptional algebras,  G2, F4, E6, E7, E8, comprise all of the simple Lie  algebras [6]. They are called Simple Classical Lie Algebras.

      Over an algebraically closed field of characteristic p > 0, all classical simple Lie algebras as well as exceptional algebras still make sense. Moreover, all classical Lie algebras, with one exception (An, if p | (n + 1), is simple modulo its one-dimensional center), remain simple for characteristics p > 3. These are still called classical Lie algebras. It should be mentioned that the modular Lie algebras of classical type are exactly the Lie algebras of simple algebraic groups and, thus, closely related to the theory of simple finite groups.

The theory of finite-dimensional Lie algebras over fields of positive characteristic p was initiated by E. Witt, N. Jacobson [5]. and H. Zasssenhaus [44].. Sometime before 1937, E. Witt came up with an example of a simple Lie algebra of dimension p, afterwards named the Witt algebra W1, which behaves completely differently from those Lie algebras we know in characteristic 0. Note, however, that in the characteristic p situation, most of the classical methods fail to work. Generally speaking, no Killing form is available, Lie's theorem on solvable Lie algebras is not true, semisimplicity of an algebra does not imply complete reducibility of its modules, Cartan subalgebras of simple algebras need not be abelian, and root lattices with respect to a Cartan subalgebra might be full vector spaces over the prime field.

About 30 years after the first appearance of nonclassical Lie algebras, A. Kostrikin and I. Shafarevich [12] constructed four infinite families of non-classical Lie algebras over algebraically closed fields of characteristics p > 5 and conjectured that every simple Lie algebra over an algebraically closed field of characteristic p > 5, which is closed under a p-power mapping (so-called restricted Lie algebras), is of classical or Cartan type. Much research was stimulated by the generalized Kostrikin-Shafarevich conjecture; i.e., that any finite-dimensional simple not necessarily restricted Lie algebra over a field of characteristic

 p > 5 is of classical or Cartan type. The occurrence of the Cartan type Lie algebras indicate that filtration methods should by very useful in the classification. In a second step, A. Kostrikin, I. Shafarevich  [13].  and V. Kac [8].  proved that a simple Lie algebra is of Cartan type, provided it admits a gradation with rather strong properties. R.L. Wilson [43] showed that a deformation of such Lie algebra, i.e., a simple filtered Lie algebra whose associated graded algebra satisfies these conditions, is of Cartan type as well.

In works by R. Block and R. Wilson [1] , and H. Strade [35], the generalized Kostrikin-Shafarevich conjecture was completely proved for the case p > 7. A major step in the Block-Wilson-Strade theory was the classification of finite-dimensional simple Lie algebras of absolute toral rank two. It was expected that completion of the toral-rank-two classification for p = 5 and p = 7, which was accomplished in a series of three papers by A. Premet and H. Strade ( [32] - [34.]), would be a major step in the classification of all the simple algebras in these characteristics. The main result of these papers is that any finite-dimensional simple Lie algebra of toral rank two is of either classical or Cartan type or is the restricted Melikyan algebra L(1,1).

           Sixty years after the first appearance of nonclassical Lie algebras, A. Premet and H. Strade announced the proof of the full Kostrikin-Safarevich conjecture (by managing the case p = 7). Moreover, for primes greater than 3, the simple Lie algebras fall into the following three categories: 

·        classical Lie algebras,

·        finite dimensional graded Cartan type Lie algebras and their deformations,

·        the Melikyan algebras  M(m, n), which constitute a single two-parametric family.

 

The algebra L(1,1) does not have analogs in characteristics p > 5 or characteristic zero. In characteristic p = 3 it is the simple 10 dimensional Kostrikin algebra L(1). It is interesting that L(1,1), being the only exceptional (not classical or Cartan type) Lie algebra over fields of characteristic p > 3, satisfies the assumptions of the Kac-Wilson ``Recognition Theorem''. Later, the construction of L(1,1) was generalized ([20] - [23.]), and a new two parameter family of modular simple Lie algebras L(n,m) of  dimension 5n+m+1 was obtained.Moreover, from results of [23]. it follows that the algebras L(n, m) could be described as 2-graded irreducible simple Lie algebras

 

           L = ÅI³ -2Li    where L0 is isomorphic to L0 = W1ÅF and dim(L-2) = 1        (1)

 

 Another distinguished feature of L(1,1) was that it is a  maximal subalgebra in the simple Lie p-algebra of  contact type K3  for p = 5  [29.]. In this context it was interesting to have a description of the maximal subalgebras of simple Lie algebras of Cartan type.

There have been several articles devoted to the characterization of properties of the Melikyan algebras:

1.     The rigidity of the Lie algebras L(m,n) under filtered deformations is established by M. Kuznetsov in [16] .

2.      A geometric realization of Melikyan algebras was obtained in [17].

3.     The automorphisms of the Melikyan algebras L(m ,n) were studied in [18].

  1. In A. Premet’s paper [31]  by  the means of a delicate analysis was established that the  well known Wilson's result about triangularblety of  Cartan subalgebras continues to hold for p = 7, but fails for p = 5. The minimal counterexample in the latter case is the Melikyan algebra L(1,1).
  2. In S. Kirillov’s paper [10] it is shown that the nilpotency class of sandwich subalgebras in Lie algebras of Cartan type and in Melikyan algebras over a field of characteristic

p = 5 is equal to 2m -1, where m is the sum of the heights of the variables.

  1.  The main result of A. Prement and H. Strade in a series of papers [32] – [39] is that any finite-dimensional simple Lie algebra of toral rank two is of either classical or Cartan type or is the restricted Melikyan algebra for p = 5.
  2. One of the topics discussed in paper by  A. I. Kostrikin and  A. S. Dzumadildaev [14] is the construction of Melikyan algebras as deformations of certain Hamiltonian Lie algebras.
  3. In a paper by S. Skryabin [39] tori in the restricted Melikian algebra L(1,1) were studied. The author determines the one-dimensional subtori in L(1,1) up to conjugation by automorphisms and describes their centralizers in L(1,1).

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Melikyan Hayk
10/9/2003