New simple modular Lie superalgebras as generalized prolongs

Over algebraically closed fields of characteristic p > 2, —prolongations of simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. We discover several new simple Lie superalgebras, serial and exceptional, including super versions of Brown and Melikyan algebras, and thus corroborate the super analog of the Kostrikin-Shafarevich conjecture. Simple Lie superalgebras with 2 × 2 Cartan matrices are classified.     

A Generalized Global Cartan Decomposition: A Basic Example

Suppose G ₁ ?  G are complex linear simple Lie groups. Let  ₁ ?  be the corresponding pair of Lie algebras. For the Killing-orthogonal  of  ₁ in  we have a vector space direct sum  =  ₁⊕ , which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G ₁) = (SL (4,?),Sp (2,?)); here  =  (4,?),  ₁ =  (2,?), and  = {X ?  | X ^? = X}, where X? X ^? is the symplectic involution. We prove that G  =  G ₁exp  ∪ i G ₁exp . The key point of the proof is to study in detail the set exp ; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n × 2n matrices X satisfying X ^? = X, which is here a natural counterpart of the standard Pfaffian.     

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.     

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.     

Axiomatic Aspects of N = 2 Vertex Superalgebras with Odd Formal Variables

We formulate the notion of “N = 2 vertex superalgebra with two odd formal variables” using a Jacobi identity with odd formal variables in which an N = 2 superconformal shift is incorporated into the usual Jacobi identity for a vertex superalgebra. It is shown that as a consequence of these axioms, the N = 2 vertex superalgebra is naturally a representation of the Lie superalgebra isomorphic to the three-dimensional algebra of superderivations with basis consisting of the usual conformal operator and the two N = 2 superconformal operators. In addition, this superconformal shift in the Jacobi identity dictates the form of the odd formal variable components of the vertex operators, and allows one to easily derive the useful formulas in the theory. The notion of N = 2 Neveu–Schwarz vertex operator superalgebra with two odd formal variables is introduced, and consequences of this notion are derived. In particular, we develop the duality properties which are necessary for a rigorous treatment of the correspondence with the underlying supergeometry. Various other formulations of the notion of N = 2 (Neveu–Schwarz) vertex (operator) superalgebra appearing in the mathematics and physics literature are discussed, and several mistakes in the literature are noted and corrected.     

Group Gradings on G 2

In this article, we describe all group gradings by a finite abelian group Γ of a simple Lie algebra of type G ₂ over an algebraically closed field F of characteristic zero.     

Embedding theorem ofZ-graded Lie superalgebras

The embedding theorem ofZ-graded Lie superalgebras is given and proved. As a subsidiary result it is proved that a transitiveZ-graded restricted lie superalgebm must be isomorphic toW(m,n, 1) if the dimension ofG _i satisfies a certain condition. Project supported by the National Natural Science Foundation of China and the Science of the University Doctoral Program of CNEC.     

On Cartan Invariants and Blocks of Zassenhaus Algebras #

In this article, we determine the Cartan invariants for Zassenhaus algebras W(1,n). This is done by reducing representations of generalized restricted Cartan type Lie algebra W(1,n) to representations of restricted Lie algebras W(1,1) and of ± b𝔰 ± b𝔩(2), and then extending Feldvoss-Nakano's argument on W(1,1) to the case W(1,n).     

Lie Superalgebras Associated with Constant Sectional Curvature

This note describes an observation connecting Riemannian manifolds of constant sectional curvature with a particular class of Lie superalgebras. Specifically, it is shown that the structural equations of a space M with constant sectional curvature, of one variety or another, nearly coincide with some identities satisfied by tensors which can be used to construct some specific families of Lie superalgebras. In particular, one obtains either osp(n,2), spl(n,2), or osp(4,2n) if the Riemannian manifold has constant curvature, constant holomorphic curvature or constant quaternion-holomorphic curvature, respectively.Mathematics Subject Classiffications (2000). 17A70, 53C29, 53C99, 57Rxx     

Three-Dimensional Topological Loops with Solvable Multiplication Groups

We prove that each 3-dimensional connected topological loop L having a solvable Lie group of dimension ≤5 as the multiplication group of L is centrally nilpotent of class 2. Moreover, we classify the solvable non-nilpotent Lie groups G which are multiplication groups for 3-dimensional simply connected topological loops L and dim G ≤ 5. These groups are direct products of proper connected Lie groups and have dimension 5. We find also the inner mapping groups of L.     

The Lie Module Function for Symmetric Groups #

Let V be a ? G-module where ? is the field of all complex numbers and G is a symmetric group. The purpose of this article is to give a method of analyzing the Lie powers L ⁿ (V ), for every positive integer n, by making use of the recent work of Bryant.     

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.     

Simple Nil Lie Algops

A Lie algop is a pair (A, L) where A is a commutative algebra and L is a Lie algebra operating on A by derivations. Faithful simple Lie algops (A, L) are of interest because the corresponding Lie algebras AL are simple—with some rare exceptions at characteristic 2. The simplicity and representation theory of Jordan Lie algops is reduced in Winter (2005b Winter , D. J. ( 2005b ). Lie algops and simple Lie algebras . Comm. Algebra 33 : 3157 – 3178 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) to the simplicity theory of nil Lie algops and the simplicity and representation theory of toral Lie algops. This paper is devoted to building the first of these two theories, the simplicity theory of nil Lie algops, as a structure theory.     

On Lie Nilpotent Rings and Cohen's Theorem

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L + rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Z _n (R), n ≥ 1, which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring ?Z _n (R) ∪ C? of R generated by the subset Z _n (R) ∪ C of R is also Lie nilpotent of index n.     

Characterizing n-Isoclinism Classes of Lie Algebras

In this article, we introduce the notion of the equivalence relation, n-isoclinism, between Lie algebras, and obtain some criterions under which Lie algebras are n-isoclinic. In particular, we show that n-isoclinic Lie algebras can be isoclinically embedded into one Lie algebra. Also, we present the notion of an n-stem Lie algebra and prove its existence within an arbitrary n-isoclinism class. In addition, similar to a result of Hekster [6 Hekster , N. S. ( 1986 ). On the structure of n-isoclinam classes of groups . J. Pure Appl. Algebra 40 : 63 – 85 .[Crossref], [Web of Science ®] , [Google Scholar]] in the group case, we characterize the n-stem Lie algebras in the n-isoclinism classes which contains at least one finitely generated Lie algebra L with dim (L ⁿ⁺¹) finite.     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.     

The Classification of Naturally Graded p-Filiform Leibniz Algebras

In the present article the classification of n-dimensional naturally graded p-filiform (1 ≤ p ≤ n ? 4) Leibniz algebras is obtained. A splitting of the set of naturally graded Leibniz algebras into the families of Lie and non Lie Leibniz algebras by means of characteristic sequences (isomorphism invariants) is proved.     

Group Algebras Satisfying a Certain Lie Identity

ABSTRACT

Let K be a field of characteristic p ≠ 2 and let G be any group. A characterization of group algebras KG satisfying the Lie identity [[x,y],[u,v],[z,t]] = 0 for all x,y,u,v,z,t ? KG is obtained.