1. Alexander Premet and the Classification of the simple modular Lie algebras

2. Ancient Times 1935 - 1945 Ernst Witt 1911 – 1991
Nathan Jacobson
Hans Zassenhaus
N.Jacobson

3. The first Lie algebra of new type found before 1937
Ernst Witt
The first Lie algebra of new type found before 1937
realized as
(p>3)

4. a Lie ring, and the associative
Nathan Jacobson
Theorem:
N.Jacobson
Bijection: intermediate fields of a purely inseparable field extension
F(c1 ,....., cn):F of degree 1
and subalgebras of DerF F(c1,.....,cn) carrying the structures of a F(c1,....,cn) - module,
a Lie ring, and the associative
p-power-mapping. (1937)

5. Definition:
A Lie algebra L is said to be restrictable, if there exists a mapping L L, x x[p], such that ad x[p] = (ad x)p for all xL. (1937)
Further results:
constructs p-analogues of the classical characteristic 0 series (1941, 1943)

6. modern notation: W(1;n)
Hans Zassenhaus
generalizes Witt‘s algebra
modern notation: W(1;n)
For n=1 set ug= (1+x)g to obtain Witt’s realization - with x=X+(Xp)
First classification theorem 1940 Let L be a simple Lie algebra with 1-dimensional toral CSA and dimensional root spaces. Then L=sl(2) or L=W(1;1).

7. Middle Ages 1944 - 1966 finding various new simple Lie algebras
obtaining isolated
classification results
deriving results on
structural features

8. Classical Lie algebras
Using a Chevalley basis one con-structs a Lie algebra over the integers and tensoring with any field F gives a Lie algebra over F . These are (modulo its center for sl(np)) the simple classical Lie algebras (including the exceptional types).
Chevalley 1956: constructs analogues to all characte-ristic 0 algebras by reduction mod p

9. Cartan type Lie algebras
Jacobson 1943:
Der F[X1,...,Xm]/(X1p-1,...,Xmp-m), iF,
modern notation if i=0: W(m;1)

10. Kostrikin-Shafarevic 1966:
define the four classes of restricted Cartan type Lie algebras
Witt W(m;1)
Special S(m;1)(1)
Hamiltonian H(2r;1)(2)
Contact K(2r+1;1)(1)
and point out their relation to the Lie algebras connected with E. Cartan's pseudogroups.

11. More general definition of the graded Cartan type Lie algebras.
(m): commutative algebra with unit element, generators xi(r), 1  i  m, r  0 and relations
characteristic-0-analogue: xi(r)=Xir/r!
((m)) the completion of (m),

12. Witt algebras:
„partial derivatives“

13. volume form
Hamiltonian form
Contact form
The second derived algebras are the simple graded Lie algebras of Cartan type.

14. General definition:
A simple filtered Lie algebra L is of Cartan type if
X(m;n)(2)  gr L  X(m;n)
for some X  {W, S, H, K}.
That means that L is a filtered deformation of a graded CTLA
Wilson 1976:
the filtered deformation can be given by a mapping
  Autc ((m)) such that
L ( X((m)) -1)  W(m;n)
=: X(m;n;  )
as filtered algebra. He needs an additional
„compatibility property“

15. Melikian algebras only in characteristic 5 Hayk Melikian 1980
The nowadays presentation is due to
M. Kuznetzov 1991:
Take the classical algebra G2, give the long root degree 0 and the short root degree -1. Just by chance because of characteristic 5 there is a prolongation of this grading
M-3⊃ …⊃M0=gl(2)⊃…⊃Ms
which terminates at finite dimensional algebras of dimension 5m+n with arbitrary m,n.

16. Kostrikin-Shafarevic Conjecture 1966:
Modern Times
Three events announce a radical change
Kostrikin-Shafarevic Conjecture 1966:
Every simple restricted Lie algebra over an algebraically closed field of characteristic p>5 is of classical or Cartan type.

17. G.B. Seligman 1967 Robert L. Wilson 1969
“Modular Lie algebras“, Springer --resumes all known results on modular Lie algebras.
Robert L. Wilson 1969
publishes his PhD thesis in which he presents the general concept of a Cartan type Lie algebra and proves that every known simple nonclassical Lie algebra is of this type (p>3).

18. Isolated classification
results (p>3)
Kostrikin 1967 / Premet 1986
L is classical if and only if there exist no sandwich elements (p>5).
Kuznetzov 1976, Weisfeiler 1984, Skryabin 1997
If L contains a solvable maximal subalgebra, then L is one of sl(2), W(1;n).

19. Benkart-Osborn If L has a CSA of dimension 1 (p>7), then L is one of sl(2), W(1;n), H(2;n;(1)) .
Wilson 1978, Premet If L has a CSA of toral rank 1, then L is one of sl(2), W(1;n), H(2;n;)(2) .
Block-Wilson If L is restricted having a 2-dimensional CSA, p>7. Then L is classical or W(2;1).
Wilson If L is restricted having a toral CSA, p>7. Then L is classical or W(m;1).

20. Basic classification results (p>3)
There are two basic classification results. In the course of the classification proof one tries to aply one of these.
Mills-Seligman 1957:
Let L have a toral CSA H such that dim [L, L-]=1 (0) and (+Fp)  (L,H) (0) Then L is classical.

21. Recognition Theorem
Let L be filtered (+technical items) and L(0)/L(1) the direct sum of classical simple algebras, gl(kp), sl(kp), pgl(kp), abelian. Then L is of classical, Cartan, or Melikian type.
Kac 1970: graded algebras Wilson 1976: filtered algebras
Benkart-Gregory-Premet 2009: substantial revision in the graded case

22. The final classification result
Theorem Let L be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic p>3. Then L is of classical, Cartan, or Melikian type.
Block – Wilson 1988 for L restricted, p>7
St – Wilson 1991 for p>7
Premet – St 2008 in general.

23. Isomorphisms no isomorphisms between algebras of different types
only canonical isomorphisms between classical algebras
isomorphism between CTLAs induces an isomorphism between the associated graded CTLAs.
Melikian algebras: isomorphism type of M(n1,n2) given by (n1,n2) and n1 n2.

24. Graded CTLAs determining n for L=X(m;n)(2):
universal p-envelope (Mil‘ner 1975)
restricted Lie algebra,
restricted subalgebra of L
Kostrikin-Shafarevic 1969 Krylyuk 1979
Kuznetzov 1989
Skryabin 1991

25. Consequences:
n is determined up to permutation
isomorphism type of W(m;n) is given
by m, n with n1... nm
Wilson 1969

26. Filtered CTLAs May regard L  W(m;n), ( gr L)(2) = X(m;n)(2).
Let
X(m;n)(2)  gr L  X(m;n)
Exists   Autc ((m)) such that
L ( X((m)) -1)  W(m;n)
=: X(m;n;  )
as filtered algebra.
„compatibility property“ Wilson 1976
May regard
L  W(m;n), ( gr L)(2) = X(m;n)(2).
Consequences:
X=S: (S)=u S , u-1du  1(m;n)
X=H: (H)  u 2(2r;n), u-1du  1(2r;n)

27. Type S(m;n;)(1) n1..... nm possible
determine Autc (m;n) - orbits of
{u S | u-1du  1(m;n)}
at most m+2 orbits
Wilson 1986

28. Type H(2r;n; )(2) {  u 2(2r;n)| u-1du  1(2r;n), d=0, 2 H }
n1.....nr , ni  ni+r for 1 i  r,
ni = nj  ni+r  nj+r for i < j  r
determine Autc (2r;n) - orbits of
{  u 2(2r;n)| u-1du  1(2r;n), d=0, 2 H }
2 different cases
Poisson algebras of R.Schafer
infinitly many orbits Kac, Skryabin 1986

29. Type K(2r+1;n;)(1) n1.....nr , ni  ni+r for 1 i  r,
ni = nj  ni+r  nj+r for i < j  r
H1,l(K(2r+1;n)(1), W(2r+1;n) / K(2r+1;n)) = 0
  = id
r, n determines the isomorphism type
Kuznetzov 1990

30. Alexander Premet and the Classification
1983 Algebraic groups associated with Lie-p-algebras of Cartan type
1986 On Cartan subalgebras of Lie-p-algebras
1986 Lie algebras without strong degeneration
1986 Inner ideals of modular Lie algebras

31. 1989 Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras
1994 A generalization of Wilson’s Theorem on Cartan Subalgebras of Simple Lie Algebras
1997 (with HS) Simple Lie Algebras of Small Characteristic: I. Sandwich Elements
(with HS) Simple Lie Algebras of Small Characteristic: II. Exceptional Roots

32. 2001 (with HS) Simple Lie Algebras of Small Characteristic: III
(with HS) Simple Lie Algebras of Small Characteristic: III. The Toral Rank 2 Case
2004 (with HS) Simple Lie Algebras of Small Characteristic: IV. Solvable and Classical Roots
2007 (with HS) Simple Lie Algebras of Small Characteristic: V. The non-Melikian case
2008 (with HS) Simple Lie Algebras of Small Characteristic: VI. Completion of the classification

33. 2006 (with HS) Classification of finite dimensional simple Lie algebras in prime charcteristics
Contemporary Mathematics, Vol 413, p

34. Meeting People: Madison, Hamburg, Manchester
September 1990 in Hamburg

35. Oberwolfach 1991
Hamburg 1996(?)

36. Madison 2000
Mt. Snowdon Wales 2001
1990

37. 2002
Chester
Manchester

38. in Manchester at Sasha’s home…
… and now in Hamburg in the Math institute 2002

39. Hamburg 2004

40. Milano 2013