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 Jordan , D. A. ( 2000 ). On the simplicity of Lie algebras of derivations of commutative algebras . J. Algebra 228 : 580 – 585 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). 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.     

Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

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.     

Simple algebras of Weyl type, II

Over a field of any characteristic, for a commutative associative algebra , and for a commutative subalgebra of , the vector space which consists of polynomials of elements in with coefficients in and which is regarded as operators on forms naturally an associative algebra. It is proved that, as an associative algebra, is simple if and only if is -simple. Suppose is -simple. Then, (a) is a free left -module; (b) as a Lie algebra, the subquotient is simple (except for one case), where is the center of . The structure of this subquotient is explicitly described. This extends the results obtained by Su and Zhao.

     

Invertible Linear Maps on Simple Lie Algebras Preserving Solvability

Let 𝔤 be a (finite-dimensional) complex simple Lie algebra of rank l. An invertible linear map ? on 𝔤 is said to preserve solvability in both directions if ?, as well as ?^?1, sends every solvable subalgebra to some solvable one. In this article, it is shown that an invertible linear map ? on 𝔤 preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism.     

Witt型单李超代数

设F是特征p〉2的域,A是F上结合的超交换的代数,D是域为F上A的超交换的导子.设A×D=A[D]为Witt型李超代数.从环论的角度得到了Witt型李超代数为单代数的充分必要条件.     

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.     

一类无限维李代数的同构与单性

探讨了一类的无限维李代数,并构造了此类李代数的理想,同构,同态,并对其性质作了探讨.     

Nil Lie p-algebras of slow growth

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [27 Petrogradsky, V. M. (2006). Examples of self-iterating Lie algebras. J. Algebra 302(2):881–886.[Crossref], [Web of Science ®] , [Google Scholar]], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [39 Shestakov, I. P., Zelmanov, E. (2008). Some examples of nil Lie algebras. J. Eur. Math. Soc. (JEMS) 10(2):391–398.[Crossref], [Web of Science ®] , [Google Scholar]]. There are a few more examples of self-similar finitely generated restricted Lie algebras with a nil p-mapping, but, as a rule, that algebras have no clear basis and require technical computations. Now we construct a family L(Ξ) of 2-generated restricted Lie algebras of slow polynomial growth with a nil p-mapping, where a field of positive characteristic is arbitrary and Ξ an infinite tuple of positive integers. Namely, GKdimL(Ξ)≤2 for all such algebras. The algebras are constructed in terms of derivations of infinite divided power algebra Ω. We also study their associative hulls A?End(Ω). Algebras L and A are ?²-graded by a multidegree in the generators. If Ξ is periodic then L(Ξ) is self-similar. As a particular case, we construct a continuum subfamily of non-isomorphic nil restricted Lie algebras L(Ξ_α), α∈?⁺, with extremely slow growth. Namely, they have Gelfand-Kirillov dimension one but the growth is not linear. For this subfamily, the associative hulls A have Gelfand-Kirillov dimension two but the growth is not quadratic. The virtue of the present examples is that they have clear monomial bases.     

单的完备Lie代数

给出了一个Ｈｅｉｓｅｎｂｅｒｇ代数与一个交换Ｌｉｅ代数的直和ｇ０的全形ｈ（ｇ０）和ｈ（ｇ０）的导子代数Ｄｅｒｈ（ｇ０）．证明了ｈ（ｇ０）不是一个完备Ｌｉｅ代数，但Ｄｅｒｈ（ｇ０）是一个单完备Ｌｉｅ代数．     

Nonlinear Maps on Simple Lie Algebras Preserving Lie Products

Let 𝔤 be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero. It is proved in this article that a bijective map ? on 𝔤 preserves Lie products if and only if it is a composition of a Lie algebra automorphism and a bijective map extended by an automorphism of the base field.     

Simple algebras of Weyl type

Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A⊗F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras. Su, Y., Zhao, K., Second cohornology group of generalized Witt type Lie algebras and certain representations, submitted to publication     

Extremal Bases for the Adjoint Representations of the Simple Lie Algebras

We construct n distinct weight bases, which we call extremal bases, for the adjoint representation of each simple Lie algebra 𝔤 of rank n: One construction for each simple root. We explicitly describe actions of the Chevalley generators on the basis elements. We show that these extremal bases are distinguished by their “supporting graphs” in three ways. (In general, the supporting graph of a weight basis for a representation of a semisimple Lie algebra is a directed graph with colored edges that describe the supports of the actions of the Chevalley generators on the elements of the basis.) We show that each extremal basis constructed is essentially the only basis with its supporting graph (i.e., each extremal basis is solitary), and that each supporting graph is a modular lattice. Each extremal basis is shown to be edge-minimizing: Its supporting graph has the minimum number of edges. The extremal bases are shown to be the only edge-minimizing as well as the only modular lattice weight bases (up to scalar multiples) for the adjoint representation of 𝔤. The supporting graph for an extremal basis is shown to be a distributive lattice if and only if the associated simple root corresponds to an end node for a “branchless” simple Lie algebra, i.e., type A, B, C, F, or G. For each extremal basis, basis elements for the Cartan subalgebra are explicitly expressed in terms of the h _i Chevalley generators.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Biderivations of finite-dimensional complex simple Lie algebras

In this paper, we prove that a biderivation of a finite-dimensional complex simple Lie algebra without the restriction of being skewsymmetric is an inner biderivation. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also obtain the forms of the linear commuting maps on the finite-dimensional complex simple Lie algebra or general linear Lie algebra.     

On a Result of Pointed Representations of finite Dimensional Simple Lie Algebras

There have been a great many of studies on the pointed representations of fi-nite-dimensional sanple Lie algebras.cf.[1][2]etc.In this paper we give a new proof of animpottant Lemma,and from this we derive our main result:Irreducible pointed modules of finite-dimesional simple Lie algebras are all Harish-Chandra modules.     

On the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras

We prove a part of the Cachazo-Douglas-Seiberg-Witten conjecture uniformly for any simple Lie algebra . The main ingredients in the proof are: Garland's result on the Lie algebra cohomology of ; Kostant's result on the `diagonal' cohomolgy of and its connection with abelian ideals in a Borel subalgebra of ; and a certain deformation of the singular cohomology of the infinite Grassmannian introduced by Belkale-Kumar.