On Isomorphisms and Invariants of Finite Dimensional Complex Filiform Leibniz Algebras

In this article, we propose an approach classifying a class of filiform Leibniz algebras. The approach is based on algebraic invariants. The method allows to classify all filiform Leibniz algebras (including filiform Lie algebras) in a given fixed dimensional case.     

On Restricted Leibniz Algebras

ABSTRACT

The role played by fields in relation to Galois Rings corresponds to semifields if the associativity is dropped, that is, if we consider Generalized Galois Rings instead of (associative) Galois rings. If S is a Galois ring and pS is the set of zero divisors in S, S* = S\ pS is known to be a finite {multiplicative} Abelian group that is cyclic if, and only if, S is a finite field, or S = ?/n? with n = 4 or n = p ^r for some odd prime p. Without associativity, S* is not a group, but a loop. The question of when this loop can be generated by a single element is addressed in this article.     

Leibniz Conformal Algebras of Rank Two

We classify all Leibniz conformal algebras of rank two.     

On the Derivations of Filiform Leibniz Algebras

The algebras of derivations of naturally graded Leibniz algebras are described. The existence of characteristically nilpotent Leibniz algebras in any dimension greater than 4 is proved.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 733–742.Original Russian Text Copyright ©2005 by B. A. Omirov.     

On Universal Central Extensions of Leibniz Algebras

We construct the endofunctor 𝔲𝔠𝔢 between the category of Leibniz algebras which assigns to a perfect Leibniz algebra its universal central extension, and we obtain the isomorphism 𝔲𝔠𝔢_Lie(𝔮_Lie) ? (𝔲𝔠𝔢_Leib(𝔮))_Lie, where 𝔮 is a perfect Leibniz algebra satisfying the condition [x, [x, y]] + [[x, y], x] = 0, for all x, y ∈ 𝔮. Moreover, we obtain several results concerning the lifting of automorphisms and derivations in a covering. We also study the relationship between the universal central extension of a semidirect product of perfect Leibniz algebras and the semidirect product of the universal central extension of both of them.     

3-Filiform Leibniz algebras of maximum length,whose naturally graded algebras are Lie algebras

In this article we present the classification of the 3-filiform Leibniz algebras of maximum length, whose associated naturally graded algebras are Lie algebras. Our main tools are a previous existence result by Cabezas and Pastor [J.M. Cabezas and E. Pastor, Naturally graded p-filiform Lie algebras in arbitrary finite dimension, J. Lie Theory 15 (2005), pp. 379–391] and the construction of appropriate homogeneous bases in the connected gradation considered. This is a continuation of the work done in Ref. [J.M. Cabezas, L.M. Camacho, and I.M. Rodríguez, On filiform and 2-filiform Leibniz algebras of maximum length, J. Lie Theory 18 (2008), pp. 335–350].     

Enveloping Algebras and Cohomology of Leibniz Pairs

We introduce the enveloping algebra for a Leibniz pair, and show that the category of modules over a Leibniz pair is isomorphic to the category of left modules over its enveloping algebra. Consequently, we show that the cohomology theory for a Leibniz pair introduced by Flato, Gerstenhaber, and Voronov can be interpreted by Ext-groups of modules over the enveloping algebra.     

Non-abelian Tensor Product of Leibniz Algebras and an Exact Sequence in Leibniz Homology

Abstract

Let 𝔪 and 𝔫 be two-sided ideals of a Leibniz algebra 𝔤 such that 𝔤 = 𝔪 + 𝔫. The goal of the paper is to achieve the exact sequence Ker(𝔪  𝔫 + 𝔫  𝔪 → 𝔤) → HL ₂(𝔤) → HL ₂(𝔤/𝔪) ⊕ HL ₂(𝔤/𝔫) → 𝔪 ∩ 𝔫/ [𝔪,𝔫] → HL ₁(𝔤) → HL ₁(𝔤/𝔪) ⊕ HL ₁(𝔤/𝔫) → 0, where HL denotes the Leibniz homology with trivial coefficients of a Leibniz algebra and denotes a non-abelian tensor product of Leibniz algebras.     

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.     

Leibniz and Hochschild homology

We construct and study the map from Leibniz homology HL_?(𝔥) of an abelian extension 𝔥 of a simple real Lie algebra 𝔤 to the Hochschild homology HH_??1(U(𝔥)) of the universal envelopping algebra U(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.     

Leibniz Homology of Extended Lie Algebras

Leibniz homology is a noncommutative homology theory for Lie algebras. In this paper, we compute low-dimensional Leibniz homology of extended Lie algebras.     

Solvable Leibniz Algebras with Heisenberg Nilradical

All solvable Lie algebras with Heisenberg nilradical have already been classified. We extend this result to a classification of solvable Leibniz algebras with Heisenberg nilradical. As an example, we show the complete classification of all real or complex Leibniz algebras whose nilradical is the 3-dimensional Heisenberg algebra.     

三维Leibniz代数的分类

Leibniz代数是比Lie代数更广泛的一类代数,它通常不满足反交换性.在这篇文章里我们确定了维数等于3的Leibniz代数的同构类.     

On Exact Courant Algebras

In this note, we will show that exact Courant algebras over a Lie algebra 𝔤 can be characterized via Leibniz 2-cocycles, and the automorphism group of a given exact Courant algebra is in a one-to-one correspondence with first Leibniz cohomology space of 𝔤.     

Second Homology Groups and Universal Coverings of Steinberg Leibniz Algebras of Small Characteristic

It is known that the second Leibniz homology group HL ₂ (𝔰𝔱𝔩 _n (R)) of the Steinberg Leibniz algebra 𝔰𝔱𝔩 _n (R) is trivial for n ≥ 5. In this article, we determine HL ₂(𝔰𝔱𝔩 _n (R)) explicitly (which are shown to be not necessarily trivial) for n = 3, 4 without any assumption on the base ring.     

A Comment on the Integration of Leibniz Algebras

In this note we point out that the definition of the universal enveloping dialgebra for a Leibniz algebra is consistent with the interpretation of a Leibniz algebra as a generalization not of a Lie algebra, but of the adjoint representation of a Lie algebra. From this point of view, the formal integration problem of Leibniz algebras is, essentially, trivial.     

On Nilpotent Properties of Leibniz Algebras

A generalization of a classical result from the theory of nilpotent Lie algebras to Leibniz algebras leads to several applications concerning the nilpotent properties both of these two types of algebras.