Helmut Strade

Simple Lie Algebras over Fields of Positive Characteristic: I. Structure Theory

The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a longstanding one. Work on this question during the last 35 years has been directed by the Kostrikin–Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin–Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block–Wilson–Strade–Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the two-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field. This first volume is devoted to preparing the ground for the classification work to be performed in the second volume. The concise presentation of the general theory underlying the subject matter and the presentation of classification results on a subclass of the simple Lie algebras for all odd primes will make this volume an invaluable source and reference for all research mathematicians and advanced graduate students in algebra.

Finitary Lie algebras

Abstract An algebra is called finitary if it consists of finite-rank transformations of a vector space. We classify finitary simple and finitary irreducible Lie algebras over an algebraically closed field of characteristic ≠2,3.

The classification of the simple modular Lie algebras II. The toral structure

Abstract Let L be a simple Lie algebra over an algebraically closed field of characteristic p > 7 and T an optimal torus in some p -envelope L p . We determine the action of T on the two-sections of L , which have been described in [St4]. We also give some new and noncomputational proofs to determine the conjugacy classes of the tori in W ( n ; 1) and of the Cartan subalgebras of W (1; n ).

Restricted Lie algebras with bounded cohomology and related classes of algebras

We determine the structure of restricted Lie algebras with bounded cohomology over arbitrary fields of prime characteristic. As a byproduct a classification of the serial restricted Lie algebras and the restricted Lie algebras of finite representation type is obtained. In addition, we derive complete information on the finite dimensional indecomposable restricted modules of these algebras over algebraically closed fields.

The classification of the simple modular lie algebras: V. algebras with Hamiltonian two-sections

We investigate the structure of simple modular Lie algebrasL over an algebraically closed field of characteristic p > 7. Every optimal torusT in some p-envelope ofL defines uniquely a subalgebraQ(L, T) ofL. We classify all L, for whichQ(L, T) ≠ L and which have a two-section of typeH (2; 1;Ф(τ))(1)

Locally finite dimensional lie algebras and their derivation algebras

The derivation algebras of all locally finite dimensional locally simple Lie algebras over a field of characteristic 0 are determined. Every locally finite dimensional Lie algebra of countable dimension is a subalgebra of the outer derivation algebra outder (ℒ) for every Lie algebra ℒ, which is the direct limit of diagonally embedded classical Lie algebras. These outer derivation algebras have dimension ℒ and are never locally finite dimensional.

Simple Lie Algebras over Fields of Positive Characteristic: III. Completion of the Classification

The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 35 years has been directed by the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. In the three-volume book, the author is assembling the proof of the Classification Theorem with explanations and references. The goal is a state-of-the-art account on the structure and classification theory of Lie algebras over fields of positive characteristic leading to the forefront of current research in this field. This first volume is devoted to preparing the ground for the classification work to be performed in the second volume.

Classifying the absolute toral rank two case

The problem of classifying the finite-dimensional simple Lie algebras over fields of characteristic p > 0 is a long-standing one. Work on this question during the last 35 years has been directed by the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed field of characteristic p > 5 a finite-dimensional restricted simple Lie algebra is classical or of Cartan type. This conjecture was proved for p > 7 by Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the general case of not necessarily restricted Lie algebras and p > 7 was announced in 1991 by Strade and Wilson and eventually proved by Strade in 1998. The final Block-Wilson-Strade-Premet Classification Theorem is a landmark result of modern mathematics and can be formulated as follows: Every finite-dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is of classical, Cartan, or Melikian type. This is the second volume by the author, presenting the state of the art of the structure and classification of Lie algebras over fields of positive characteristic, an important topic in algebra. The contents is leading to the forefront of current research in this field.