By Helmut Strade

The challenge of classifying the finite dimensional basic Lie algebras over fields of attribute p > zero is a long-standing one. paintings in this query has been directed by means of the Kostrikin-Shafarevich Conjecture of 1966, which states that over an algebraically closed box of attribute p > five a finite dimensional constrained uncomplicated Lie algebra is classical or of Cartan variety. This conjecture used to be proved for p > 7 via Block and Wilson in 1988. The generalization of the Kostrikin-Shafarevich Conjecture for the final case of no longer inevitably constrained Lie algebras and p > 7 used to be introduced in 1991 by way of Strade and Wilson and at last proved through Strade in 1998. the ultimate Block-Wilson-Strade-Premet class Theorem is a landmark results of smooth arithmetic and will be formulated as follows: *Every uncomplicated finite dimensional uncomplicated Lie algebra* *over an algebraically closed box of attribute p > three is of classical, Cartan, or Melikian type.*

In the three-volume publication, the writer is assembling the facts of the type Theorem with factors and references. The objective is a state of the art account at the constitution and category concept of Lie algebras over fields of optimistic attribute.

This first quantity is dedicated to getting ready the floor for the class paintings to be played within the moment and 3rd volumes. The concise presentation of the overall concept underlying the subject material and the presentation of class effects on a subclass of the straightforward Lie algebras for all strange primes will make this quantity a useful resource and reference for all study mathematicians and complex graduate scholars in algebra. the second one version is corrected.

**Contents**

Toral subalgebras in *p*-envelopes

Lie algebras of precise derivations

Derivation easy algebras and modules

Simple Lie algebras

Recognition theorems

The isomorphism problem

Structure of straightforward Lie algebras

Pairings of brought about modules

Toral rank 1 Lie algebras