Lie-admissible Algebras Research Articles

Products of Commutator Ideals of Some Lie-admissible Algebras

Products of Commutator Ideals of Some Lie-admissible Algebras

On the free metabelian Novikov and metabelian Lie-admissible algebras

On the free metabelian Novikov and metabelian Lie-admissible algebras

An Effective Criterion for Nielsen–Schreier Varieties

Abstract To the memory of V. A. Artamonov (1946–2021) All algebras of a certain type are said to form a Nielsen–Schreier variety if every subalgebra of a free algebra is free. This property has been perceived as extremely rare; in particular, only six Nielsen–Schreier varieties of algebras with one binary operation have been discovered in prior work on this topic. We propose an effective combinatorial criterion for the Nielsen–Schreier property in the case of algebras over a field of zero characteristic; in our approach, operads play a crucial role. Using this criterion, we show that the well-known varieties of all pre-Lie algebras and of all Lie-admissible algebras are Nielsen–Schreier, and, quite surprisingly, that there are already infinitely many non-equivalent Nielsen–Schreier varieties of algebras with one binary operation and identities of degree four.

Certain lie algebraic structures on Riemannian manifolds with semi-symmetric non-metric connection

As a natural consequence of the Levi-Civita connection on a Riemannian manifold, there is a Lie algebra structure on a Riemannian manifold. Lie Algebras and Lie Groups are the mathematical structure of continuous symmetries in physics. In this paper, semi-symmetric non-metric connection is considered instead of Levi-Civita connection of Riemann manifold, and accordingly the existence of algebraic structures is investigated. First, it is shown that there is not always a Lie algebra structure on a Riemannian manifold with a semi-symmetric non-metric connection. Then, necessary and sufficient conditions for Lie admissible algebra, pre-Lie algebra and post Lie algebra on a Riemann manifold with semi-symmetric non-metric connection are obtained depending on geometric terms. In addition, the cases of the Riemannian manifold with such algebraic structures according to the semi-symmetric non-metric connection being Einstein manifold and being flat manifold have been also investigated.

Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebras

We introduce the notion of anti-pre-Lie algebras as the underlying algebraic structures of nondegenerate commutative 2-cocycles which are the “symmetric” version of symplectic forms on Lie algebras. They can be characterized as a class of Lie-admissible algebras whose negative left multiplication operators make representations of the commutator Lie algebras. We observe that there is a clear analogy between anti-pre-Lie algebras and pre-Lie algebras by comparing them in terms of several aspects. Furthermore, it is unexpected that a subclass of anti-pre-Lie algebras, namely admissible Novikov algebras, correspond to Novikov algebras in terms of q-algebras. Consequently, there is a construction of admissible Novikov algebras from commutative associative algebras with derivations or more generally, admissible pairs. The correspondence extends to the level of Poisson type structures, leading to the introduction of the notions of anti-pre-Lie Poisson algebras and admissible Novikov-Poisson algebras, whereas the latter correspond to Novikov-Poisson algebras.

Free Lie-admissible algebras and an analogue of the PBW theorem

We first construct linear bases for free Lie-admissible algebras and develop a new theory of Gröbner–Shirshov bases for Lie-admissible algebras. Then we prove an analogue of the Poincaré–Birkhoff–Witt theorem, that is, every Lie algebra L can be embedded into its universal enveloping Lie-admissible algebra U(L), where the basis of U(L) does not depend on the multiplication table of L. Finally, we show that the basic rank of the variety of Lie-admissible algebras is 1. As a corollary, the universal enveloping Lie-admissible algebra of an abelian Lie algebra does not satisfy any nontrivial identity in the variety of Lie-admissible algebras.

Algebraic Structure and Poisson Integral Method of Snake-Like Robot Systems

The algebraic structure and Poisson's integral of snake-like robot systems are studied. The generalized momentum, Hamiltonian function, generalized Hamilton canonical equations, and their contravariant algebraic forms are obtained for snake-like robot systems. The Lie-admissible algebra structures of the snake-like robot systems are proved and partial Poisson integral methods are applied to the snake-like robot systems. The first integral methods of the snake-like robot systems are given. An example is given to illustrate the results.

Weakly associative algebras, Poisson algebras and deformation quantization

Weakly associative algebras are a class of Lie-admissible algebras that generalize associative algebras and useful to extend the class of formal deformations of associative commutative algebras. Thus we obtain a new method of determining quantization deformations of Poisson algebras which generalizes the classical process. The algebraic study of weakly associative algebras is also interesting, in particular the homology of free algebras with one generator that coincides with the free flexible algebra with one generator.

Lie-Admissible Algebras Associated with Dynamical Systems

We introduce the general structures of Lie-admissible algebras in the spaces of Gâteaux differentiable operators and establish their connection with the symmetries of operator equations and the mechanics of infinite-dimensional systems.

Zinbiel algebras and bialgebras: main properties and related algebraic structures

This work provides a characterization of left and right Zinbiel algebras. Basic identities are established and discussed, showing that Zinbiel algebras are center-symmetric, and therefore Lie-admissible algebras. Their bimodules are given, and used to build a Zinbiel algebra structure on the direct sum of the underlying vector space and a finite-dimensional vector space. In addition, their matched pair is built, and related to the matched pair of their sub-adjacent Lie algebras. Besides, Zinbiel coalgebras are introduced, and linked to their underlying Lie coalgebras and coassociative coalgebras. Moreover, the related Manin triple is defined, and used to characterize Zinbiel bialgebras, and their equivalence to the associated matched pair.

Construction of $\Gamma$-algebra and $\Gamma$-Lie admissible algebras

In this paper, at first we generalize the notion of algebra over a field. A $\Gamma$-algebra is an algebraic structure consisting of a vector space $V$, a groupoid $\Gamma$ together with a map from $V\times\Gamma\times V$ to $V$. Then, on every associative $\Gamma$-algebra $V$ and for every $\alpha\in \Gamma$ we construct an $\alpha$-Lie algebra. Also, we discuss some properties about $\Gamma$-Lie algebras when $V$ and $\Gamma$ are the sets of $m\times n$ and $n\times m$ matrices over a field $F$ respectively. Finally, we define the notions of $\alpha$-derivation, $\alpha$-representation, $\alpha$-nilpotency and prove Engel theorem in this case.

A Class of Lie-admissible Algebras

A Class of Lie-admissible Algebras

Lie-admissible algebras associated with dynamical systems

Lie-admissible algebras associated with dynamical systems

Construction of dialgebras through bimodules over algebras

The varieties of dialgebras (also known as Loday-type algebras) over a given type of algebra have been the subject of multiple recent developments. We provide here a construction of such dialgebra varieties via bimodules over an algebra and a surjective equivariant map. Our construction is equivalent to the KP construction (Kolesnikov–Pozhidaev construction) when departing from the set of linearized identities of the algebra variety. The novel construction simplifies the obtention of the dialgebra equations without forcing a complete linearization of the algebra identities. We illustrate the use of the novel construction providing the dialgebras associated to several varieties of algebras, including those over diverse Lie admissible algebras. We provide some novel explorations on the structure of the dialgebras which are easily articulated through our construction.

A Class of Nonassociative Algebras Including Flexible and Alternative Algebras, Operads and Deformations

There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group Σ3. The first one corresponds to the Lie-admissible algebras and this class has been studied in a previous paper of Remm and Goze. Here we are interested by the second one corresponding to the third power associative algebras.

AN ALGEBRA WITH RIGHT IDENTITIES AND ITS ANTISYMMETRIZED ALGEBRA

We define the Lie-admissible algebra NW<TEX>$({\mathbb{F}}[e^{A[s]},x_1,{\cdots},x_n])$</TEX> in this work. We show that the algebra and its antisymmetrized (i.e., Lie) algebra are simple. We also find all the derivations of the algebra NW<TEX>$(F[e^{{\pm}x^r},x])$</TEX> and its antisymmetrized algebra W<TEX>$(F[e^{{\pm}x^r},x])$</TEX> in the paper.

Hom-algebra structures

Hom-algebra structures are given on linear spaces by multiplications twisted by linear maps. Hom-Lie algebras and general quasi-Hom-Lie and quasi-Lie algebras were introduced by Hartwig, Larsson and Silvestrov as algebras embracing Lie algebras, super and color Lie algebras and their quasi-deformations by twisted derivations. In this paper we introduce and study Hom-associative, Hom-Leibniz and Hom-Lie admissible algebraic structures generalizing associative, Leibniz and Lie admissible algebras. Also, we characterize flexible Hom-algebras and explain some connections and differences between Hom-Lie algebras and Santilli’s isotopies of associative and Lie algebras.

Algebras with one operation including Poisson and other Lie-admissible algebras

We investigate algebras with one operation. We study when these algebras form a monoidal category and analyze Koszulness and cyclicity of the corresponding operads. We also introduce a new kind of symmetry for operads, the dihedrality, responsible for the existence of dihedral cohomology. The main trick, which we call the polarization, will be used to represent an algebra with one operation without any specific symmetry as an algebra with one commutative and one anticommutative operations. We will try to convince the reader that this change of perspective might sometimes lead to new insights and results. This point of view was used by Livernet and Loday to introduce a one-parameter family of operads whose specialization at 0 is the operad for Poisson algebras, while at a generic point it equals the operad for associative algebras. We study this family and explain how it can be used to interpret the deformation quantization (∗-product) in a neat and elegant way.

Lie-admissible algebras and operads

A Lie-admissible algebra gives a Lie algebra by anticommutativity. In this work we describe remarkable types of Lie-admissible algebras such as Vinberg algebras, pre-Lie algebras or Lie algebras. We compute the corresponding binary quadratic operads and study their duality. Considering Lie algebras as Lie-admissible algebras, we can define for each Lie algebra a cohomology with values in an Lie-admissible module. This leads to the study some deformations of Lie algebras in the classes of Lie-admissible algebras.

Opérades Lie-admissibles

The aim of this paper is to present remarkable classes of Lie-admissible algebras containing in particular the associative algebras, the Vinberg and pre-Lie algebras. We determine the associated operads and their dual operads. To cite this article: E. Remm, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1047–1050.