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A non-abelian tensor product of Leibniz algebra

Allahtan Victor Gnedbaye (1999)

Annales de l'institut Fourier

Leibniz algebras are a non-commutative version of usual Lie algebras. We introduce a notion of (pre)crossed Leibniz algebra which is a simultaneous generalization of notions of representation and two-sided ideal of a Leibniz algebra. We construct the Leibniz algebra of biderivations on crossed Leibniz algebras and we define a non-abelian tensor product of Leibniz algebras. These two notions are adjoint to each other. A (co)homological characterization of these new algebraic objects enables us to...

Cartan subalgebras, weight spaces, and criterion of solvability of finite dimensional Leibniz algebras.

Sergio A. Albeverio, Ayupov, Shavkat, A. 2, Bakhrom A. Omirov (2006)

Revista Matemática Complutense

In this work the properties of Cartan subalgebras and weight spaces of finite dimensional Lie algebras are extended to the case of Leibniz algebras. Namely, the relation between Cartan subalgebras and regular elements are described, also an analogue of Cartan s criterion of solvability is proved.

Classification of p-adic 6-dimensional filiform Leibniz algebras by solutions of x q = a

Manuel Ladra, Bakhrom Omirov, Utkir Rozikov (2013)

Open Mathematics

We study the p-adic equation x q = a over the field of p-adic numbers. We construct an algorithm which gives a solvability criteria in the case of q = p m and present a computer program to compute the criteria for any fixed value of m ≤ p − 1. Moreover, using this solvability criteria for q = 2; 3; 4; 5; 6, we classify p-adic 6-dimensional filiform Leibniz algebras.

Growth of some varieties of Leibniz-Poisson algebras

Ratseev, S. M. (2011)

Serdica Mathematical Journal

2010 Mathematics Subject Classification: 17A32, 17B63.Let V be a variety of Leibniz-Poisson algebras over an arbitrary field whose ideal of identities contains the identities {{x1,y1},{x2,y2},ј,{xm,ym}} = 0, {x1,y1}·{x2,y2}· ј ·{xm,ym} = 0 for some m. It is shown that the exponent of V exists and is an integer.

Left-right noncommutative Poisson algebras

José Casas, Tamar Datuashvili, Manuel Ladra (2014)

Open Mathematics

The notions of left-right noncommutative Poisson algebra (NPlr-algebra) and left-right algebra with bracket AWBlr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NPlr-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWBlr and NPlr-algebras the notions of actions, representations, centers, actors and crossed modules are described as...

Leibniz cohomology for differentiable manifolds

Jerry M. Lodder (1998)

Annales de l'institut Fourier

We propose a definition of Leibniz cohomology, H L * , for differentiable manifolds. Then H L * becomes a non-commutative version of Gelfand-Fuks cohomology. The calculations of H L * ( R n ; R ) reduce to those of formal vector fields, and can be identified with certain invariants of foliations.

Opérades différentielles graduées sur les simplexes et les permutoèdres

Frédéric Chapoton (2002)

Bulletin de la Société Mathématique de France

On définit plusieurs opérades différentielles graduées, dont certaines en relation avec des familles de polytopes : les simplexes et les permutoèdres. On obtient également une présentation de l’opérade K liée aux associaèdres introduite dans un article antérieur.

Sets with two associative operations

Teimuraz Pirashvili (2003)

Open Mathematics

In this paper we consider duplexes, which are sets with two associative binary operations. Dimonoids in the sense of Loday are examples of duplexes. The set of all permutations carries a structure of a duplex. Our main result asserts that it is a free duplex with an explicitly described set of generators. The proof uses a construction of the free duplex with one generator by planary trees.

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