Metric n-Lie Algebras

An n-Lie algebra is said to be metric if it is endowed with an invariant, non-degenerate, symmetric bilinear form. We prove that any simple n-Lie algebra over an algebraically closed field of characteristic zero admits a unique metric structure and vice versa. Further, we present two metric n-Lie algebras, which are indecomposable but admit many more metric structures.     

T*-extension of a 3-Lie algebra

We introduce a new technique called T*-extension, of constructing a metric 3-Lie algebra out of an arbitrary 3-Lie algebra, and explore all possible metrics and corresponding signatures on this resulting metric 3-Lie algebra.     

3-李代数的辛结构

研究了3-李代数和度量3-李代数的辛结构.对任意3-李代数L,构造了无限多个度量辛3-李代数.证明了度量3-李代数(A,B)是度量辛3-李代数的充要条件,即存在可逆导子D,使得D∈Der_B(A).同时证明了每一个度量辛3-李代数(A,B,ω)是度量辛3-李代数(A,B,ω)的T_θ~*-扩张.最后,利用度量辛3-李代数经过特殊导子的双扩张得到了新的度量辛3-李代数.     

Nilpotent n-Lie Algebras

We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I ² nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent.     

On Schur’s theorem and its converses for n-Lie algebras

An n-Lie algebra analogue of Schur’s theorem and its converse as well as a Lie algebra analogue of Baer’s theorem and its converse are presented. Also, it is shown that, an n-Lie algebra with finite dimensional derived subalgebra and finitely generated central factor is isoclinic to some finite dimensional n-Lie algebra.     

n-Ary Mal'tsev Algebras

By analogy with n-Lie algebras, which are a natural generalization of Lie algebras to the case of n-ary multiplication, we define the concept of an n-ary Mal'tsev algerba. It is shown that exceptional algebras of a vector cross product are ternary central simple Mal'tsev algebras, which are not 3-Lie algebras if the characteristic of a ground field is distinct from 2 and 3. The basic result is that every n-ary algebra of the vector cross product is an n-ary central simple Mal'tsev algebra.     

T *-extension of n-Lie algebras

We introduce a new technique, called the T?*-extension, of constructing metric n-Lie algebras from arbitrary ones and obtain some important properties on this extension. Finally, we investigate the metric n-Lie algebras that are isometric to certain T?*-extensions.     

A class of non-nilpotent 2-solvable n-Lie algebras

This article concerns a class of finite-dimensional minimal non-nilpotent 2-solvable n-Lie algebras. It is shown that if L is a finite-dimensional minimal non-nilpotent 2-solvable n-Lie algebra, then L can be decomposed into a semi-direct of an ideal A and an (n ? 1)-dimensional subalgebra H ₀ of L. Furthermore, H ₀ acts irreducibly on A/A ¹, and H ₀ + A ¹ is a self-normalizing maximal subalgebra of L with the core A ¹, the derived algebra of A.     

Some Properties On Isoclinism in n-Lie Algebras

The concept of n-Lie algebra were introduced by Filippov in 1987. One notes that not all properties of Lie algebras can be carried over to n-Lie algebras, as Williams showed in 2009. In the present article, among other results it is shown that the notions of isoclinism and isomorphism for two finite dimensional n-Lie algebras of the same dimension are equivalent. This was already done for ordinary Lie algebras by Moneyhun in 1994.     

Universal Associative Envelopes of (n+ 1)-Dimensional n-Lie Algebras

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie (Filippov) algebras. More generally, for n even and any (n + 1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map L → U(L) is injective. We use noncommutative Gröbner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.     

The n-lie property of the Jacobian as a condition for complete integrability

We prove that an associative commutative algebra U with derivations D ₁, ..., D _n ε DerU is an n-Lie algebra with respect to the n-multiplication D ₁ ^ ? ^ D _n if the system {D ₁, ..., D _n } is in involution. In the case of pairwise commuting derivations this fact was established by V. T. Filippov. One more formulation of the Frobenius condition for complete integrability is obtained in terms of n-Lie multiplications. A differential system {D ₁, ..., D _n } of rank n on a manifold M ^m is in involution if and only if the space of smooth functions on M is an n-Lie algebra with respect to the Jacobian Det(D _i u _j ).     

Extending involutions on Frobenius algebras

 Let A be a central simple algebra of degree n over a field of characteristic different from 2 and let B ? A be a maximal commutative subalgebra. We show that if there is an involution on A that preserves B and such that the socle of each local component of B is a homogeneous C ₂ -module for this action, then B is a Frobenius algebra. For a fixed commutative Frobenius algebra B of finite dimension n equipped with an involution σ, we characterize the central simple algebras A of degree n that contain B and carry involutions extending σ. Received: 29 October 2001 / Revised version: 2 February 2002     

Cartan Subalgebras of Leibniz n-Algebras

The present article is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz n-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz n-algebra and Cartan subalgebras and regular elements of the corresponding factor n-Lie algebra is established.     

Additive Lie (ξ-Lie) Derivations and Generalized Lie (ξ-Lie) Derivations on Prime Algebras

The additive(generalized)ξ-Lie derivations on prime algebras are characterized. It is shown, under some suitable assumptions, that an additive map L is an additive generalized Lie derivation if and only if it is the sum of an additive generalized derivation and an additive map from the algebra into its center vanishing all commutators; is an additive(generalized)ξ-Lie derivation with ξ = 1 if and only if it is an additive(generalized)derivation satisfying L(ξA)= ξL(A)for all A. These results are then used to characterize additive(generalized)ξ-Lie derivations on several operator algebras such as Banach space standard operator algebras and von Neumman algebras.     

Irreducible highest weight representations of the simple n-Lie algebra

A. Dzhumadil??daev classified all irreducible finite dimensional representations of the simple n-Lie algebra. Using a slightly different approach, we obtain in this paper a complete classification of all irreducible, highest weight modules, including the infinite-dimensional ones. As a corollary we find all primitive ideals of the universal enveloping algebra of this simple n-Lie algebra.     

(H,R)-Lie coalgebras and (H,R)-Lie bialgebras

Given an (H,R)-Lie coalgebra Γ, we construct (H,R _T )-Lie coalgebra Γ^T through a right cocycle T, where (H,R) is a triangular Hopf algebra, and prove that there exists a bijection between the set of (H,R)-Lie coalgebras and the set of ordinary Lie coalgebras. We also show that if (L, [, ], Δ, R) is an (H,R)-Lie bialgebra of an ordinary Lie algebra then (L ^T , [, ], Δ_T, R _T ) is an (H,R _T )-Lie bialgebra of an ordinary Lie algebra.     

Derivations,automorphisms, and representations of complex ω-Lie algebras

Let (𝔤,ω) be a finite-dimensional non-Lie complex ω-Lie algebra. We study the derivation algebra Der(𝔤) and the automorphism group Aut(𝔤) of (𝔤,ω). We introduce the notions of ω-derivations and ω-automorphisms of (𝔤,ω) which naturally preserve the bilinear form ω. We show that the set Der_ω(𝔤) of all ω-derivations is a Lie subalgebra of Der(𝔤) and the set Aut_ω(𝔤) of all ω-automorphisms is a subgroup of Aut(𝔤). For any three-dimensional and four-dimensional nontrivial ω-Lie algebra 𝔤, we compute Der(𝔤) and Aut(𝔤) explicitly, and study some Lie group properties of Aut(𝔤). We also study representation theory of ω-Lie algebras. We show that all three-dimensional nontrivial ω-Lie algebras are multiplicative, as well as we provide a four-dimensional example of ω-Lie algebra that is not multiplicative. Finally, we show that any irreducible representation of the simple ω-Lie algebra C_α(α≠0,?1) is one-dimensional.     

On the structure of A/k-bialgebras in prime characteristic

In this paper, we study A/k-bialgebras in prime characteristic. Firstly, we prove a Cartier type structure theorem for cocomplete A/k-bialgebras. Secondly, we generalize Michaelis’s theorem. In particular, we prove that a restricted (k, K)-Lie algebra is proper if and only if its restricted enveloping algebra is proper.     

The Lower Socle and Finite Rank Elementary Operators

Abstract

We define the lower socle of a semiprime algebra 𝒜 as the sum of all minimal left ideals 𝒜e where e is a minimal idempotent such that the division algebra e𝒜e is finite dimensional. We study the connection between the condition that the elements a _k , b _k , 1 ≤ k ≤ n, lie in the lower socle of 𝒜 and the condition that the elementary operator x ? a ₁ xb ₁ + ? + a _n xb _n has finite rank. As an application we obtain some results on derivations certain of whose powers have finite rank.