Formulas in the Free Lie Algebras

A formula in the free Lie algebra implies a simple proof of two classical characterizations of Lie elements.     

Automorphisms of Relatively Free Nilpotent Lie Algebras

Let L be a relatively free nilpotent Lie algebra over ? of rank n and class c, with n ≥ 2; freely generated by a set 𝒵. Give L the structure of a group, denoted by R, by means of the Baker–Campbell–Hausdorff formula. Let G be the subgroup of R generated by the set 𝒵 and N _Aut(L)(G) the normalizer in Aut(L) of the set G. We prove that the automorphism group of L is generated by GL _n (?) and N _Aut(L)(G). Let H be a subgroup of finite index in Aut(G) generated by the tame automorphisms and a finite subset X of IA-automorphisms with cardinal s. We construct a set Y consisting of s + 1 IA-automorphisms of L such that Aut(L) is generated by GL _n (?) and Y. We apply this particular method to construct generating sets for the automorphism groups of certain relatively free nilpotent Lie algebras.     

Test Sets in Free Metabelian Lie Algebras

We calculate the test rank of a finite rank free metabelian Lie algebra over an arbitrary field and characterize the test sets for these algebras. We prove that each automorphism that is the identity modulo the derived subalgebra and that acts as the identity on some test set is an inner automorphism.     

Asymptotic Behavior in Lie Nilpotent Relatively Free Algebras and Extended Grassmann Algebras

The present paper is devoted to estimating the codimensions c_n for the variety of associative algebras given by the identity [x₁, …, x_l] = 0 for odd l. The characteristic of the ground field is different from 2 and 3. The construction of the so-called extended Grassmann algebra is applied very effectively.

     

Determination of Endomorphisms of Free Metabelian Lie Algebras

Siberian Mathematical Journal -     

Ideals of Free Metabelian Lie Algebras and Primitive Elements

Necessary and sufficient conditions are established for elements of a finite rank free metabelian Lie algebra in order that every endomorphism of this algebra be uniquely determined by its values at these elements.     

Free Kac–Moody Groups and their Lie Algebras

We compute the Lie algebras of continuous right invariant derivations of free Kac–Moody groups.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Zero Divisors in Amalgamated Free Products of Lie Algebras

We introduce the notion of domain for Lie algebras, discuss when the free amalgamated Lie product of two domains is a domain again, classify commuting elements in the free amalgamated product of two Lie algebras and describe the center of this product.     

Endomorphisms of Free Metabelian Lie Algebras Which Preserve Orbits

For the free rank 2 metabelian Lie algebra over an infinite field we prove that an endomorphism of the algebra which preserves the automorphic orbit of a nonzero element is an automorphism. We construct some counterexamples over finite fields.     

简约李代数和局部李代数的生成元

证明了具有n-维中心的简约李代数的生成元的最小个数是n或n+1(n≥1),也证明了满足一定约束条件的局部李代数能由两个元素生成.     

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.     

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.     

关于完备李群与完备李代数

孟道骥等对完备李代数作了系统的研究并已获得很多基本和重要的结果。本文给出完备李群与完备李代数的某些关系。     

Polynomial Lie Algebras

We introduce and study a special class of infinite-dimensional Lie algebras that are finite-dimensional modules over a ring of polynomials. The Lie algebras of this class are said to be polynomial. Some classification results are obtained. An associative co-algebra structure on the rings k[x ₁,...,x _n]/(f ₁,...,f _n) is introduced and, on its basis, an explicit expression for convolution matrices of invariants for isolated singularities of functions is found. The structure polynomials of moving frames defined by convolution matrices are constructed for simple singularities of the types A,B,C, D, and E ₆.