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.     

Prime Quotients of Jordan Systems and Lie Algebras

We show that, unlike alternative algebras, prime quotients of a nondegenerate Jordan system or a Lie algebra need not be nondegenerate, even if the original Jordan system is primitive, or the Lie algebra is strongly prime, both with nonzero simple hearts. Nevertheless, for Jordan systems and Lie algebras directly linked to associative systems, we prove that even semiprime quotients are necessarily nondegenerate.     

Generalised Chain Conditions,Prime Ideals,and Classes of Locally Finite Lie Algebras

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals.Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras.We study conditions on prime ideals relating these properties.We prove that the radicalof any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals,and an ideally finite Lie algebra is quasi-Noetherian if and only if it is qussi-Artinian.Both properties are equivalent to soluble-by-finite.We also prove a structure theorem for serially finite Artinian Lie algebras.     

Naturally reductive metrics on homogeneous systems

We show that naturally reductive (indefinite) metrics on homogeneous systems are determined by nondegenerate invariant forms of their tangent Lie triple algebras. By using this we obtain the decomposition theorem of homogeneous systems. Furthermore, we show that a naturally reductive Riemannian homogeneous space is irreducible if and only if its tangent Lie triple algebra is simple.     

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.     

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H²_Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.     

The isomorphic realization of nondegenerate solvable Lie algebras of maximal rank based on Kac-Moody algebras

In this paper,based on Kac-Moody algebra,the isomorphic realization of nondegenerate solvable Lie algebras of maximal rank is given,which in turn revels the closed connections between nondegenerate solvable Lie algebras and Kac-Moody algebras,resulting in some new worthy topics in this area.     

The degeneracy of extended affine Lie algebras

We construct degenerate extended affine Lie algebras from a given nondegenerate extended affine Lie algebra and show that all degenerate extended affine Lie algebras are obtained in this way. Received: 21 January 1997     

Solvable quadratic Lie algebras

A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.     

Abelian Lie symmetry algebras of two-dimensional quasilinear evolution equations

We carry out the classification of abelian Lie symmetry algebras of two-dimensional second-order nondegenerate quasilinear evolution equations. It is shown that such an equation is linearizable if it admits an abelian Lie symmetry algebra that is of dimension greater than or equal to 5 or of dimension greater than or equal to 3 with rank-one.     

算子李代数的一些性质

算子群作为群的推广,算子群在群论里有许多应用.类似地,作为算子群和李代数的推广,算子李代数将会有许多应用.给出了算子李代数的一些性质,得到了算子李代数半单性的充分必要条件.同时得到算子李代数半单性与非退化killing型的关系.     

Integrable and non-integrable structures in Einstein-Maxwell equations with Abelian isometry group <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

Using the fact that absolute zero divisors in Jordan pairs become Lie sandwiches of the corresponding Tits–Kantor–Koecher Lie algebras, we prove local nilpotency of the McCrimmon radical of a Jordan system (algebra, triple system, or pair) over an arbitrary ring of scalars. As an application, we show that simple Jordan systems are always nondegenerate.     

三次可解型非退化李代数及其导子李代数的结构

本文给出了非退化可解李代数的两个类型:三次可解型非退化李代数和扩充的 Heisenberg李代数,并确定三次可解型非退化李代数及其导子李代数的结构．     

Lie algebras with an algebraic adjoint representation revisited

A well-known theorem due to E. Zelmanov proves that PI-Lie algebras with an algebraic adjoint representation over a field of characteristic zero are locally finite-dimensional. In particular, a Lie algebra (over a field of characteristic zero) whose adjoint representation is algebraic of bounded degree is locally finite-dimensional. In this paper it is proved that a prime nondegenerate PI-Lie algebra with an algebraic adjoint representation over a field of characteristic zero is simple and finite-dimensional over its centroid, which is an algebraic field extension of the base field. We also give a new and shorter proof of the local finiteness of Lie algebras with an algebraic adjoint representation of bounded degree.     

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.     

POINTED REPRESENTATIONS OFINFINITE DIMENSIONAL LIE ALGEBRAS

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On the Variations of $G_2%

In [1], Shen Guangyu constructed several classes of new simple Lie algebras of characteristic 2, which are called the variations of $G_2$. In this paper, the authors investigate their derivation algebras. It is shown that $G_2$ and its variations all possess unique nondegenerate associative forms. The authors also find some nonsingular derivations of $V_iG$ for $i=3,4,5,6,$ and thereby construct some left-symmetric structures on $V_iG$ for $i=3,4,5,6.$ Some errors about the variations of $sl(3,F)$ in [1] are corrected.     

ON THE VARIATIONS OF G_2

In [1], Shen Guangyu constructed several classes of new simple Lie algebras of characteristic 2, which are called the variations of G2. In this paper, the authors investigate their derivation algebras. It is shown that G2 and its variations all possess unique nondegenerate associative forms. The authors also find some nonsingular derivations of ViG for i = 3,4, 5, 6, and thereby construct some left-symmetric structures on Vi G for i = 3,4,5,6. Some errors about the variations of sl(3, F) in [1] are corrected.     

Non-degenerate Invariant (Super)Symmetric Bilinear Forms on Simple Lie (Super)Algebras

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac–Moody (super)algebras are the most known examples.     

THE INVARIANT FORMS ON THE GRADED MODULES OF THE GRADED CARTAN TYPE LIE ALGEBRAS

This paper determines the graded modules of graded Cartan type Lie aigebras whichpossess nondegenerate invariant form.