Alexander Premet and the Classification of the simple

Alexander Premet and the Classification of the simple modular Lie algebras

Ancient Times 1935 - 1945 Ø Ernst Witt Ø Nathan Ø Hans 1911 – 1991 Jacobson 1910 - 1999 Zassenhaus 1912 - 1991 2

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

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

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) 5

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 1 -dimensional root spaces. Then L=sl(2) or L=W(1; 1). 6

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

Classical Lie algebras Using a Chevalley basis one constructs 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 characteristic 0 algebras by reduction mod p 8

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

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

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), 11

Witt algebras: „partial derivatives“ 12

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

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 -1 L ( X((m)) ) W(m; n) =: X(m; n; ) as filtered algebra. He needs an additional „compatibility property“ 14

Melikian algebras only in characteristic 5 Hayk Melikian 1980 The nowadays presentation is due to M. Kuznetzov 1991: Take the classical algebra G 2, 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⊃ …⊃M 0=gl(2)⊃…⊃Ms which terminates at finite dimensional algebras of dimension 5 m+n with arbitrary 15 m, n.

Modern Times 1966 - 1992 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. 16

G. B. Seligman 1967 “Modular Lie algebras“, Springer -resumes all known results on modular Lie algebras. Robert L. Wilson 1969 publishes his Ph. D 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). 17

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). 18

Benkart-Osborn 1984 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 1994 If L has a CSA of toral rank 1, then L (2) is one of sl(2), W(1; n), H(2; n; ). Block-Wilson 1982 If L is restricted having a 2 dimensional CSA, p>7. Then L is classical or W(2; 1). Wilson 1983 If L is restricted having a toral CSA, p>7. Then L is classical or W(m; 1). 19

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. 20

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: Wilson 1976: graded algebras filtered algebras Benkart-Gregory-Premet 2009: substantial revision in the graded case 21

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. 22

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(n 1, n 2) given by (n 1, n 2) and n 1 n 2. 23

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 24

Consequences: • n is determined up to permutation • isomorphism type of W(m; n) is given by m, n with n 1. . . nm Wilson 1969 25

Filtered CTLAs 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-1 du 1(m; n) • X=H: ( H) u 2(2 r; n), u-1 du 1(2 r; n) 26

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

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

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

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 30

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 1999 (with HS) Simple Lie Algebras of Small Characteristic: II. Exceptional Roots 31

2001 (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 32

2006 (with HS) Classification of finite dimensional simple Lie algebras in prime charcteristics Contemporary Mathematics, Vol 413, p. 185 - 214 33

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

Oberwolfach 1991 Hamburg 1996(? ) 35

Madison 2000 Mt. Snowdon Wales 2001 1990 36

2002 Chester Manchester 37

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

Hamburg 2004 39

Milano 2013 40

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