A Survey on Single-Valued Neutrosophic $K$-Algebras

In this survey, we first present a brief overview of logical algebras. We then discuss concepts of single-valued neutrosophic $K$-subalgebras, single-valued neutrosophic soft $K$-algebras and single-valued neutrosophic topological $K$-algebras. Moreover, we discuss various fundamental concepts which include interior, closure, $C{_5}$-connectivity, super connectivity, compactness and Hausdorffness of single-valued neutrosophic topological $K$-algebras.     

Heisenberg Lie Color Algebras

In this article, we give a sufficient condition for a Lie color algebra to be complete. The color derivation algebra Der(?) and the holomorph L of finite dimensional Heisenberg Lie color algebra ? graded by a torsion-free abelian group over an algebraically closed field of characteristic zero are determined. We prove that Der(?) and Der(L) are simple complete Lie color algebras, but L is not a complete Lie color algebra.     

Embedding of Lie Algebras in Isogeneralized Structurable Algebras

In a previous paper, the first-named author introduced generalized structurable algebras, while the second-named author introduced the isotopies of Lie algebras. In this paper, we combine the two analyses, submit the notion of isogeneralized structural algebras, and show that they include Lie algebras, all their axiom-preserving generalizations of graded, supersymmetric or isotopic type, as well as numerous other algebras.     

Flat Nonunimodular Lorentzian Lie Algebras

A flat Lorentzian Lie algebra is a left symmetric algebra endowed with a symmetric bilinear form of signature (?, +,…, +) such that left multiplications are skew-symmetric. In geometrical terms, a flat Lorentzian Lie algebra is the Lie algebra of a Lie group with a left-invariant Lorentzian metric with vanishing curvature. In this article, we show that any flat nonunimodular Lorentzian Lie algebras can be obtained as a double extension of flat Riemannian Lie algebras. As an application, we give all flat nonunimodular Lorentzian Lie algebras up to dimension 4.     

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.     

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.     

Triple Homomorphisms of Perfect Lie Algebras

Let L, L′ be Lie algebras over a commutative ring R. A R-linear mapping f: L → L′ is called a triple homomorphism from L to L′ if f([x, [y, z]]) = [f(x), [f(y), f(z)]] for all x, y, z ∈ L. It is clear that homomorphisms, anti-homomorphisms, and sums of homomorphisms and anti-homomorphisms are all triple homomorphisms. We proved that, under certain assumptions, these are all triple homomorphisms.     

The Isomorphism Problem for Universal Enveloping Algebras of Lie Algebras

Let L be a Lie algebra with universal enveloping algebra U(L). We prove that if H is another Lie algebra with the property that U(L) ≅ U(H) then certain invariants of L are inherited by H. For example, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H. Presented by D. Passman     

Group Algebras of Finite Groups as Lie Algebras

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group G, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of G.     

Quadratic Lie Algebras

In this article, the notion of universal enveloping algebra introduced in Ardizzoni [4 Ardizzoni , A. On primitively generated braided bialgebras . Algebr. Represent. Theory , to appear. (doi:10.1007/s10468-010-9257-z)  [Google Scholar]] is specialized to the case of braided vector spaces whose Nichols algebra is quadratic as an algebra. In this setting, a classification of universal enveloping algebras for braided vector spaces of dimension not greater than 2 is handled. As an application, we investigate the structure of primitively generated connected braided bialgebras whose braided vector space of primitive elements forms a Nichols algebra, which is a quadratic algebra.     

Derivation Simple Color Algebras and Semisimple Lie Color Algebras

Let K be an algebraically closed field of arbitrary characteristic and Γ an abelian multiplicative group equipped with a bicharacter ε: Γ × Γ → K*. It is proved that, for any finite-dimensional derivation simple color algebra A over K, there exists a simple color algebra S and a color vector space V such that A? S? S^ε(V), where S^ε(V) is the ε-symmetric algebra of V. As an application of this result, a necessary and sufficient condition such that a Lie color algebra is semisimple is obtained.     

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.     

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.     

A Generalization of Extended Affine Lie Algebras

We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples.     

Lie isomorphisms of reflexive algebras

A Lie isomorphism ? between algebras is called trivial if ?=ψ+τ, where ψ is an (algebraic) isomorphism or a negative of an (algebraic) anti-isomorphism, and τ is a linear map with image in the center vanishing on each commutator. In this paper, we investigate the conditions for the triviality of Lie isomorphisms from reflexive algebras with completely distributive and commutative lattices (CDCSL). In particular, we prove that a Lie isomorphism between irreducible CDCSL algebras is trivial if and only if it preserves I-idempotent operators (the sum of an idempotent and a scalar multiple of the identity) in both directions. We also prove the triviality of each Lie isomorphism from a CDCSL algebra onto a CSL algebra which has a comparable invariant projection with rank and corank not one. Some examples of Lie isomorphisms are presented to show the sharpness of the conditions.     

整环上二阶线性李代数的自同构

设Ｒ是有１的交换环，２是Ｒ的单位．本文决定了Ｒ上李代数ｓｌ２（Ｒ）的理想．进而，若Ｒ是整环，本文决定了ｓｌ２（Ｒ）与ｇｌ２（Ｒ）的自同构形式．     

Lie Nilpotent Group Algebras and Upper Lie Codimension Subgroups

In this article we introduce the series of the upper Lie codimension subgroups of a group algebra KG of a group G over a field K. By means of this series we give a contribution to the conjecture cl _L (KG) = cl ^L (KG) when G belongs to particular classes of finite p-groups.     

Multiplicative Semiprimeness of Skew Lie Algebras

Abstract

Let A be a semiprime associative algebra with an involution over a field of characteristic not 2, let K be the Lie algebra of all skew elements of A, and let Z _{[K, K]} denote the annihilator of the Lie algebra [K, K]. We will prove that the multiplication algebra of the semiprime Lie algebra [K, K]/Z _{[K, K]} is also semiprime. As a consequence, the multiplication algebra of [K, K]/Z _{[K, K]} is prime, whenever [K, K]/Z _{[K, K]} is prime. We will obtain similar results for the Lie algebra K/Z _K whenever the base field has characteristic zero.