Leibniz Conformal Algebras of Rank Two

We classify all Leibniz conformal algebras of rank two.     

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.     

Test Sets and Test Rank of a Free Metabelian Lie Algebra

Abstract

We prove that the test rank of a free metabelian Lie algebra L of rank n equals n-1 and a test set of L belongs to the derived subalgebra of L.     

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.     

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.     

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.     

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.     

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.     

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.     

An Example of Constructing Versal Deformation for Leibniz Algebras

In this work we compute a versal deformation of the 3-dimensional nilpotent Leibniz algebra over ?, defined by the nontrivial brackets [e ₁, e ₃] = e ₂ and [e ₃, e ₃] = e ₁.     

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.     

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.     

Leibniz Central Extensions on Some Infinite-Dimensional Lie Algebras

Abstract

In this paper, all one-dimensional Leibniz central extensions on the algebras of differential operators over C[t, t ^?1] and C((t)), as well as on the quantum 2-torus, the Virasoro-like algebra and its q-analog are studied. We determine all nontrivial Leibniz 2-cocycles on these infinite dimensional Lie algebras.     

Leibniz代数上的粗糙集

把粗糙集理论方法应用到Leibniz代数上,定义了Leibniz代数上的同余关系,给出了Leibniz代数的粗糙子代数和粗糙理想等概念,研究了Leibniz代数上粗糙集在同态映射下的若干性质.     

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points

ABSTRACT

Let F _m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F _m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F _m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) ^G is not finitely generated.     

Leibniz Algebras with Invariant Bilinear Forms and Related Lie Algebras

In ([11 Benayadi, S., Hidri, S. (2014). Quadratic Leibniz algebras. Journal of Lie Theory 24:737–759.[Web of Science ®] , [Google Scholar]]), we have studied quadratic Leibniz algebras that are Leibniz algebras endowed with symmetric, nondegenerate, and associative (or invariant) bilinear forms. The nonanticommutativity of the Leibniz product gives rise to other types of invariance for a bilinear form defined on a Leibniz algebra: the left invariance, the right invariance. In this article, we study the structure of Leibniz algebras endowed with nondegenerate, symmetric, and left (resp. right) invariant bilinear forms. In particular, the existence of such a bilinear form on a Leibniz algebra 𝔏 gives rise to a new algebra structure ☆ on the underlying vector space 𝔏. In this article, we study this new algebra, and we give information on the structure of this type of algebras by using some extensions introduced in [11 Benayadi, S., Hidri, S. (2014). Quadratic Leibniz algebras. Journal of Lie Theory 24:737–759.[Web of Science ®] , [Google Scholar]]. In particular, we improve the results obtained in [22 Lin, J., Chen, Z. (2010). Leibniz algebras with pseudo-Riemannian bilinear forms. Front. Math. China 5(1):103–115.[Crossref], [Web of Science ®] , [Google Scholar]].     

Free Akivis Algebras, Primitive Elements, and Hyperalgebras

Free Akivis algebras and primitive elements in their universal enveloping algebras are investigated. It is proved that subalgebras of free Akivis algebras are free and that finitely generated subalgebras are finitely residual. Decidability of the word problem for the variety of Akivis algebras is also proved.The conjecture of K. H. Hofmann and K. Strambach (Problem 6.15 in [Topological and analytic loops, in “Quasigroups and Loops Theory and Applications,” Series in Pure Mathematics (O. Chein, H. O. Pflugfelder, and J. D. H. Smith, Eds.), Vol. 8, pp. 205–262, Heldermann Verlag, Berlin, 1990]) on the structure of primitive elements is proved to be not valid, and a full system of primitive elements in free nonassociative algebra is constructed.Finally, it is proved that every algebra B can be considered as a hyperalgebra, that is, a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop, where the hyperalgebra operations on B are interpreted by certain primitive elements.     

Endomorphisms of Free Metabelian Lie Algebras Which Preserve Orbits

For the free rank 2 metabelian Lie algebra over an infinite field we prove that an endomorphism of the algebra which preserves the automorphic orbit of a nonzero element is an automorphism. We construct some counterexamples over finite fields.