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.     

Algebraic Lie algebras with complete Borel subalgebras

In this article we prove that an algebraic Lie algebra over an algebraically closed field of characteristic 0 is complete if its Borel subalgebras are complete. Thus the study on complete Lie algebras may somewhat be reduced to that on solvable complete ones.     

量子环面上一类导子李代数的结构和自同构群

本文研究量子环面上的一类导子李代数,它包含了Virasoro-Like代数及其q类似．首先证明了这 类导子李代数之间的同构一定是分次同构,并进一步给出了代数同构的充要条件及同构映射的具体表达 式,最后确定了该类李代数的自同构群．     

von Neumann代数中套子代数上的Lie同构

本文给出因子von Neumann代数中的幂等算子在广义Lie积下的一个刻画; 得到因子von Neumann代数中套子代数的幂等算子在Lie积下的一个特征．作为应用, 研究了因子von Neumann代数中套子代数上的Lie同构,并证明因子von Neumann 代数中套子代数之间的Lie同构,要么是同构与广义迹之和,要么是负反同构与广义迹之和．     

Leibniz algebras constructed by Witt algebras

ABSTRACT

We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.     

Assosymmetric algebras under Jordan product

We prove that assosymmetric algebras under the Jordan product are Lie triple algebras. A Lie triple algebra is called special if it is isomorphic to a subalgebra of the plus-algebra of some assosymmetric algebra. We establish that the Glennie identity of degree 8 is valid for special Lie triple algebras, but not for all Lie triple algebras.     

Lie Bialgebras of Generalized WITT Type,II

In an article by Michaelis, a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In a recent article by Song and Su, Lie bialgebra structures on graded Lie algebras of generalized Witt type with finite dimensional homogeneous components were considered. In this article we consider Lie bialgebra structures on the graded Lie algebras of generalized Witt type with infinite dimensional homogeneous components. By proving that the first cohomology group H¹(𝒲, 𝒲 ? 𝒲) is trivial for any graded Lie algebras 𝒲 of generalized Witt type with infinite dimensional homogeneous components, we obtain that all such Lie bialgebras are triangular coboundary.     

Contact and Frobenius solvable Lie algebras with abelian nilradical

The aim of this work is to characterize the families of Frobenius (respectively, contact) solvable Lie algebras that satisfies the following condition: 𝔤 = 𝔥?V, where 𝔥?𝔤𝔩(V), |dim V?dim 𝔤|≤1 and NilRad(𝔤) = V, V being a finite dimensional vector space. In particular, it is proved that every complex Frobenius solvable Lie algebra is decomposable, whereas that in the real case there are only two indecomposable Frobenius solvable Lie algebras.     

Classification of three-dimensional zeropotent algebras over the real number field

A nonassociative algebra is defined to be zeropotent if the square of any element is zero. In this paper, we give a complete classification of three-dimensional zeropotent algebras over the real number field up to isomorphism. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional real Lie algebras, which is in accordance with the Bianchi classification. Moreover, three-dimensional zeropotent algebras over a real closed field are classified in the same manner as those over the real number field.     

The Core of a Locally Extended Affine Lie Algebra

Locally extended affine Lie algebras are a general version of extended affine Lie algebras. In this article, we completely describe the structure of the core of a locally extended affine Lie algebra. We prove that the core of a locally extended affine Lie algebra is a direct limit of Lie tori.     

Generalized cartan type k lie algebras in characteristic 0

The Lie algebra of Cartan type K which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra F[x₀, x₁,…, x_n,x_n?1,…,x_?n], where F is a field of characteristic 0, was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials F[x₀,x₁,…, x_n,x_?1,…,x_?n,X₀ ^?1x₁ ^-1,…,x_n ^?1,…,x_?1 ^?1…,x_?n ^?1]A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, determine all possible     

BRST Operator for Quantum Lie Algebras: Relation to the Bar Complex

Quantum Lie algebras (an important class of quadratic algebras arising in the Woronowicz calculus on quantum groups) are generalizations of Lie (super)algebras. Many notions from the theory of Lie (super)algebras admit quantum generalizations. In particular, there is a BRST operator Q (Q ²=0) that generates the differential in the Woronowicz theory and gives information about (co)homologies of quantum Lie algebras. In our previous papers, we gave and solved a recursive relation for the operator Q for quantum Lie algebras. Here, we consider the bar complex for q-Lie algebras and its subcomplex of q-antisymmetric chains. We establish a chain map (which is an isomorphism) of the standard complex for a q-Lie algebra to the subcomplex of the antisymmetric chains. The construction requires a set of nontrivial identities in the group algebra of the braid group. We also discuss a generalization of the standard complex to the case where a q-Lie algebra is equipped with a grading operator.     

On Lie nilpotent modular group algebras

Let K be a field of characteristic p>0 and let KG be the group algebra of an arbitrary group G over K. It is known that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is at least p+1. The group algebras KG for which these indices are p+1 or 2p or 3p?1 or 4p?2 have already been determined. In this paper, we classify the group algebras KG for which the upper Lie nilpotency index is 5p?3, 6p?4 or 7p?5.     

On the duality of generalized Lie and Hopf algebras

We show how, under certain conditions, an adjoint pair of braided monoidal functors can be lifted to an adjoint pair between categories of Hopf algebras. This leads us to an abstract version of Michaelis' theorem, stating that given a Hopf algebra H  , there is a natural isomorphism of Lie algebras Q(H)^?≅P(H^°)$Q{(H)}^{?}\cong P(H^{°})$, where Q(H)^?$Q{(H)}^{?}$ is the dual Lie algebra of the Lie coalgebra of indecomposables of H  , and P(H^°)$P(H^{°})$ is the Lie algebra of primitive elements of the Sweedler dual of H. We apply our theory to Turaev's Hopf group-(co)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.     

On Anosov automorphisms of nilmanifolds

The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds, and the existence of such automorphisms is a really strong condition on the rational nilpotent Lie algebra determined by the lattice, so called an Anosov Lie algebra. We prove that n⊕?⊕n (s times, s≥2) has an Anosov rational form for any graded real nilpotent Lie algebra n having a rational form. We also obtain some obstructions for the types of nilpotent Lie algebras allowed, and use the fact that the eigenvalues of the automorphism are algebraic integers (even units) to show that the types (5,3) and (3,3,2) are not possible for Anosov Lie algebras.