David J. Winter

Symmetric Lie algebras

Lie Rootsysrems R are introduced, with axioms which reflect properties of the rootset of a Lie algebra L as structured by representations of compatible simple restricted rank I subquotients of L. The rank 1 Lie rootsystems and the rank 2 Lie rootsystems defined over H, are classified up to isomorphism. Base. closure and core are discussed. The rootsystems of collapse on passage from R to Core R are shown to be of type S,,. Given any Lie rootsystem R, its independent root pairs are shown to fall into eleven classes. Where the eleventh (anomoly) pair Tz never occurs, it is shown that R is contained in R, + S (not always equal), where R, is a Witr rootsystem and S is a classical roolsystem. This result is of major importance to two papers (D. J. Winter, Generalized classicallAlberttZassenhaus Lie algebras, to appear; Rootsystems of simple Lie algebras, to appear), since it implies that the rootsystems of the simple nonclassical Lie algebras considered there are Witt rootsystems. Toral Lie algebras and swmetric Lie algehrus are introduced and studied as generalizations of classical-Albert&Zassenhaus Lie algebras. It is shown that their rootsystems are Lie rootsystems. The cores of toral Lie algebras are shown to be classical-Albert-Zassenhaus Lie algebras. These results form the basis for the abovementioned papers on rootsystems of simple Lie algebras and the classification of the rootsystems of two larger classes of Lie algebras, the generalized classical-Albert-Zassenhaus Lie algebras and the classical-AlbertPZassenhaus-Kaplansky Lie algebras. Symmetric Lie algebras are introduced as generalizations of classicallAlbertZassenhaus Lie algebras. It is shown that their rootsets R are Lie rootsystems. Consequently, symmetric Lie algebras can be studied locally using the classification of rank 2 Lie rootsystems. This is done in detail for [oraL Lie algebras. i“ 1985 Academsc

Lie Algops and Simple Lie Algebras

ABSTRACT Pairs (A, L) with A a commutative algebra and L a Lie algebra acting on A by derivations, called Lie algops, are studied as algebraic structures over arbitrary fields of arbitrary characteristic. Lie algops possess modules and tensor products—and are considered with respect to a central simple theory. The simplicity problem of determining the faithful unital simple Lie algops ( A, L ) is of interest since the corresponding Lie algebras AL are usually simple (Jordan, 2000). For locally finite Lie algops, and up to purely inseparable descent, this problem reduces by way of closures to the closed central simplicity problem of determining those which are closed central simple. The simplicity and representation theories for locally nilpotent separably triangulable unital Lie algops are of particular interest because they relate to the problems of classifying simple Lie algebras of Witt type and their representations. Of these, the simplicity theory reduces to that of Jordan Lie algops. The main Theorems 7.3 and 7.4 reduce the simplicity and representation theories for Jordan Lie algops to the simplicity and representation theories for simple nil and toral Lie algops.

Simple Nil Lie Algops

A Lie algop is a pair (A, L) where A is a commutative algebra and L is a Lie algebra operating on A by derivations. Faithful simple Lie algops (A, L) are of interest because the corresponding Lie algebras AL are simple—with some rare exceptions at characteristic 2. The simplicity and representation theory of Jordan Lie algops is reduced in Winter (2005b) to the simplicity theory of nil Lie algops and the simplicity and representation theory of toral Lie algops. This paper is devoted to building the first of these two theories, the simplicity theory of nil Lie algops, as a structure theory.

Rootsystems of simple Lie algebras

W),S,(S,).Witt rootsystems having no sections Sz. We ( W v W) are classified as those rootsystems whose irreducible components are finite vector space subgroups. Since the latter are rootsystems of generalized Albert-Zassenhaus Lie algebras, it follows that the rootsystems of nonclassical simple Lie algebras L =x,,sR L, such that ([e,f]) # 0 for some e E Lb, f~ L’-, for each a E R - (0) which contain no section of type T2, S,, or W@ ( W v W) are classified up to isomorphism by finite vector

Simple Lie Algebras

Abstract Finite and infinite dimensional simple Lie algebras S(A, L) are constructed from a commutative algebra A and nil locally nilpotent Lie subalgebra L of the derivation algebra of A under conditions which are met when A is a domain of characteristic not 2.

Generalized classical Albert-Zassenhaus Lie algebras

Two very large classes GCAZ and CAZK of Lie algebras are introduced, which contain all sums of classical, Albert-Zassenhaus, generalized Witt algebras of Kaplansky and associated holomorphs. Their rootsystems R are classitied up to isomorphism. The group Aut L of automorphisms of L is shown to contain extensions of the Weyl group of R and the inner automorphism groups of classical Lie algebra complements of the Wilt subalgebra of L. The Weyl group extension in Aut L

Central Simple Nonassociative Algebras with Operators

The classical central simple theory of associative algebras generalizes, in this article, to a central simple theory of nonassociative algebras with operators and a related central irreducible theory of modules. These theories are motivated by, and apply to, problems of constructing and classifying simple Jordan Lie algebras, irreducible modules, and birings.

The structure of symmetric Lie algebras

In Winter [7], a certain class of Lie algebras, s~*mmrtric Lie ab and then we use the theory of algebraic groups and the theory of Lie rootsystems to pro\.p the following theorem, which expresses the structure of a symmetric Lie aigebra L in terms of a classica! Lie algebra L,S, a semisimple symmetric Lie algebra Lw whose root system is a Witt rootsystem (defined below) and solvable ideals. In the case of a ground field of characteristic 0, the theorem simply says that a symmetric Lie algebra of characteristic 0 is of the form L = L, @ Solv L with L, semisimple, which follows from Levi’s Theorem. So. we restrict ourselves in this paper to the much more difficult context of a ground field k of prime characteristic p > 3 (and sometimes 7).

Biring Theory and Its Galois Theory of Rings

Biring theory is about birings (A, P), that is, algops (A, P) of an associative algebra A and (A, A)-biring P acting on A via a morphism γ: P → Pres F A from P to the terminal (A, A)-biring Pres F A of preservations of A. (The word biring is used in a theory for a structure with unit, product, counit, coproduct subject to conditions of the theory.) Biring theory has its central simple theory and its Galois theory of rings. Its Galois birings are the reduced simple birings.

A Lie algebra variation on a theorem of Wedderburn

THEOREM. Let A be a finite dimensional associative algebra over a field F. Suppose that A has a basis over F consisting of nilpotent elements. Then A itself must be nilpotent. Given a finite dimensional Lie algebra L which is a Lie subalgebra of an associative algebra A (possibly infinite dimensional), we can ask what results from supposing that L has a basis over F consisting of elements that are nilpotent in the associative algebra. We cannot conclude that L is nilpotent and finite dimensional. In fact, this is rarely the case. For example,