The socle of a nondegenerate Lie algebra

We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.     

Invariant Lie Algebras and Lie Algebras with a Small Centroid

A subalgebra of a Lie algebra is said to be invariant if it is invariant under the action of some Cartan subalgebra of that algebra. A known theorem of Melville says that a nilpotent invariant subalgebra of a finite-dimensional semisimple complex Lie algebra has a small centroid. The notion of a Lie algebra with small centroid extends to a class of all finite-dimensional algebras. For finite-dimensional algebras of zero characteristic with semisimple derivations in a sufficiently broad class, their centroid is proved small. As a consequence, it turns out that every invariant subalgebra of a finite-dimensional reductive Lie algebra over an arbitrary definition field of zero characteristic has a small centroid.     

Trialitarian automorphisms of lie algebras

Over an algebraically closed field of characteristic zero simple Lie algebras admit outer automorphisms of order 3 if and only if they are of type D₄. Moreover, thereare two conjugacy classes of such automorphisms. Among orthogonal Lie algebras over arbitrary fields of characteristic zero, only orthogonal Lie algebras relative to quadratic norm forms of Cayley algebras admit outer automorphisms of order 3. We give a complete list of conjugacy classes of outer automorphisms of order 3 for orthogonal Lie algebras over arbitrary fields of characteristic zero. For the norm form of a given Cayley algebra, one class is associated with the Cayley algebra and the others with central simple algebras of degree 3 with involution of the second kind such that the cohomological invariant of the involution is the norm form.     

An Artinian theory for Lie algebras

Complemented Lie algebras are introduced in this paper (a notion similar to that studied by O. Loos and E. Neher in Jordan pairs). We prove that a Lie algebra is complemented if and only if it is a direct sum of simple nondegenerate Artinian Lie algebras. Moreover, we classify simple nondegenerate Artinian Lie algebras over a field of characteristic 0 or greater than 7, and describe the Lie inner ideal structure of simple Lie algebras arising from simple associative algebras with nonzero socle.     

The Jordan socle and finitary Lie algebras

In this paper we introduce the notion of Jordan socle for nondegenerate Lie algebras, which extends the definition of socle given in [A. Fernández López et al., 3-Graded Lie algebras with Jordan finiteness conditions, Comm. Algebra, in press] for 3-graded Lie algebras. Any nondegenerate Lie algebra with essential Jordan socle is an essential subdirect product of strongly prime ones having nonzero Jordan socle. These last algebras are described, up to exceptional cases, in terms of simple Lie algebras of finite rank operators and their algebras of derivations. When working with Lie algebras which are infinite dimensional over an algebraically closed field of characteristic 0, the exceptions disappear and the algebras of derivations are computed.     

Left-symmetric algebras of derivations of free algebras

A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric algebra of derivations of a free nonassociative symmetric m-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described.     

Lie algebras with almost dimensionally nilpotent inner derivations

The structure of the Lie algebras with almost dimensionally nilpotent inner derivations is studied. It is proved that, if the base field is of characteristic 0, then, when d>6 is odd, there exist just two d-dimensional Lie algebras; when d>6 is even, there exists just one d-dimensional Lie algebra such that these Lie algebras are nonsolvable and have some almost dimensionally nilpotent inner derivations.     

Invariant Subspaces of Operator Lie Algebras and the Theory of K-Algebras

It is proved that if a Lie algebra of compact operators contains a nonzero ideal consisting of quasinilpotent operators then this Lie algebra has a nontrivial invariant subspace. Some applications of this result to lattices of invariant subspaces for families of compact operators and to structures of ideals of Banach Lie algebras with compact adjoint action are given.     

A Characterisation of Nilpotent Lie Algebras by Invertible Leibniz-Derivations

Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques. We only consider finite-dimensional Lie algebras over a fields of characteristic zero.     

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

The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials . 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. Received December 20, 1996; in final form September 15, 1997     

A remark on the liftable derivation of moduli algebras of isolated hypersurface singularities

An example is given to show that not every derivation in the nilradical of the Lie algebra of derivations of moduli algebras can be liftable and the dimension of the nilradical of the Lie algebra of derivations of moduli algebras is not a topological invariant for an isolated hypersurface singularity.

     

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. Dedicated to Prof. H. Petersson on the occasion of his 60th birthday     

单扩张型Lie Rinehart代数的分类定理

定义单扩张型Lie Rinehart代数,从而给出一种通过导子构造Lie Rinehart代数的途径.指出这是一种特殊的作用Lie Rinehart代数.在系数环是没有零因子的交换代数的前提下,给出单扩张型Lie Rinehart代数的完全分类定理.特别的,证明多项式环上的任何非平凡作用Lie Rinehart代数必然是单扩张型的,并给出其标准型.     

Left-symmetric algebras and homogeneous improper affine spheres

The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the nth power of the function. The exponentials of the characteristic polynomials of certain left-symmetric algebras yield examples of such functions whose level sets are analogues of the generalized Cayley hypersurface of Eastwood–Ezhov. There are found purely algebraic conditions sufficient for the characteristic polynomial of the left-symmetric algebra to have the desired properties. Precisely, it suffices that the algebra has triangularizable left multiplication operators and the trace of the right multiplication is a Koszul form for which right multiplication by the dual idempotent is projection along its kernel, which equals the derived Lie subalgebra of the left-symmetric algebra.     

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (?/p ?)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L ₀ = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.     

Ternary Derivations of Generalized Cayley–Dickson Algebras

Abstract

Ternary derivations, ternary Cayley derivations and ternary automorphisms are computed over fields of characteristic ≠ 2, 3 for the algebras A _t obtained by the Cayley–Dickson duplication process. While the derivation algebra of A _t stops growing after t = 3, the ternary derivation algebra significantly decreases in the step from the octonions A ₃ to the sedenions A ₄, revealing the symmetry lost on that stage.     

Lie Derivations of Triangular Algebras

We investigate Lie derivations on a class of algebras called triangular algebras. In particular, we give sufficient conditions such that every Lie derivation on such an algebra is a sum of derivation on and a mapping from to its centre.     

Derivations of the Even Part of Finite-Dimensional Simple Modular Lie Superalgebra M

Let F be the underlying base field of characteristic p > 3 and denote by M the even part of the finite-dimensional simple modular Lie superalgebra M. In this paper, the generator sets of the Lie algebra M which will be heavily used to consider the derivation algebra Der(M) are given. Furthermore, the derivation algebra of M is determined by reducing derivations and a torus of M, i.e.,As a result, the derivation algebra of the even part of M does not equal the even part of the derivation superalgebra of M.     

Constructions for Octonion and Exceptional Jordan Algebras

In this note we reverse theusual process of constructing the Lie algebras of types G ₂and F ₄ as algebras of derivations of the splitoctonions or the exceptional Jordan algebra and instead beginwith their Dynkin diagrams and then construct the algebras togetherwith an action of the Lie algebras and associated Chevalley groups.This is shown to be a variation on a general construction ofall standard modules for simple Lie algebras and it is well suitedfor use in computational algebra systems. All the structure constantswhich occur are integral and hence the construction specialisesto all fields, without restriction on the characteristic, avoidingthe usual problems with characteristics 2 and 3.     

Stochastic integration for set-indexed processes

Diagonalizable derivations of a finite-dimensional algebra usually span an ideal in the Lie algebra of all derivations. This ideal is studied for underlying graded, monomial, and path algebras.