J. Marshall Osborn

Flexible Lie-admissible algebras☆

Abstract This paper investigates finite-dimensional flexible Lie-admissible algebras A over fields of characteristic 0. Under these hypotheses the vector space A with the Lie product[ x , y ] = xy − yx is a Lie algebra, denoted by A − . The main result of this work gives a characterization of those flexible Lie-admissible algebras for which the solvable radical of A − is a direct summand of A − . Included in this class of algebras are all flexible Lie-admissible A for which A − is a reductive Lie algebra. Our technique is to view A as a module for a certain semisimple Lie algebra of derivations of A and to see what restrictions the module structure imposes on the multiplication of A . A subsequent investigation will show that this module approach can also be used to determine the flexible Lie-admissible algebras A for which the radical of A − is abelian.

Identities of non-associative algebras

In the first part of this paper we define a partial ordering on the set of all homogeneous identities and find necessary and sufficient conditions that an identity does not imply any identity lower than it in the partial ordering (we call such an identity irreducible). Perhaps the most interesting property established for irreducible identities is that they are skew-symmetric in any two variables of the same odd degree and symmetric in any two variables of the same even degree. The results of the first section are applied to commutative algebras in the remainder of the paper. It is proved that any commutative algebra with unity element of characteristic not 2, 3, or 5 satisfying an identity of degree 4 or less not implied by the commutative law is either power-associative or satisfies one of two other identities. A similar, but more complicated theorem is proved for commutative algebras satisfying identities of degree 5. An application of the results of §1 to non-commutative algebras has already been made in (1).

Rank one Lie algebras

Let L denote a finite-dimensional Lie algebra over an algebraically closed field (. Such an algebra is called a rank one Lie algebra provided L has a one-dimensional Cartan subalgebra. When (D is of characteristic zero, a fundamental result in the theory of simple Lie algebras over (D is that the only rank one algebra is the 3-dimensional Lie algebra B t(2) of traceless 2 x 2 matrices. The representation theory of t t(2) plays a critical role in the structure and classification of simple algebras over ( as well as in the study of their representations. When the field (D is of prime characteristic p > 3, in addition to B [ (2), the Albert-Zassenhaus Lie algebras ([1]; see also [3]) are known to be rank one algebras. By an Albert-Zassenhaus Lie algebra is meant an algebra over (D with basis { gala E G } where G is a finite additive subgroup of (D, and with multiplication defined by

Nonassociative algebras related to Hamiltonian operators in the formal calculus of variations

Abstract The main result in this paper is the classification of simple Novikov algebras A with a maximal subalgebra H such that A H has a finite-dimensional irreducible H-submodule. A second result deals with the extension of Hamiltonian operators.

Generalized Poisson brackets and lie algebras of type H in characteristic 0

Abstract. The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra

Generalized cartan type k lie algebras in characteristic 0

F[x_1,\dots,x_n, x_{-1},\dots,x_{-n}]

Toral rank one Lie algebras

was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials

Contributions to the classification of simple modular Lie algebras

F[x_1,\dots,x_n,x_{-1}, \dots,x_{-n}, x_1^{-1},\dots,x_n^{-1},x_{-1}^{-1},\dots,x_{-n}^{-1}]

Infinite Dimensional Lie Algebras of Type L

. We show in this paper that these generalizations of Cartan type H algebras are isomorphic to certain generalizations of the classical algebra of Poisson brackets, and that it can be generalized further. In turn, these algebras can be recast in a form that is an adaption of a class of Lie algebras of characteristic p that was defined in 1958 be R. Block. A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, find their derivations, and determine all possible isomorphisms between two of these algebras.

Simple Lie Algebras

The Lie algebra of Cartan type K which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra F[x0, x1,…, xn,xn−1,…,x−n], where F is a field of characteristic 0, was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials F[x0,x1,…, xn,x−1,…,x−n,X0 −1x1 -1,…,xn −1,…,x−1 −1…,x−n −1]A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, determine all possible