Zinbiel Algebras Research Articles

Some Results on Zinbiel Algebras and Rota–Baxter Operators

Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, ZAs are also realized in low dimensions of the RBOs of commutative associative algebras. It was found that not all ZAs can be attained in this way.

Twisted Lie algebras by invertible derivations

In this paper, we introduce an algebra structure denoted by InvDer algebra in which we twist an algebra with an invertible derivation whose inverse is also a derivation. We define InvDer Lie algebras, InvDer associative algebras, InvDer Zinbiel algebras and InvDer dendriforme algebras. We also study the relations between these structures by using Rota–Baxter operators and endomorphism operators.

On Pre-Novikov Algebras and Derived Zinbiel Variety

For a non-associative algebra $A$ with a derivation $d$, its derived algebra $A^{(d)}$ is the same space equipped with new operations $a\succ b = d(a)b$, $a\prec b = ad(b)$, $a,b\in A$. Given a variety ${\rm Var}$ of algebras, its derived variety is generated by all derived algebras $A^{(d)}$ for all $A$ in ${\rm Var}$ and for all derivations $d$ of $A$. The same terminology is applied to binary operads governing varieties of non-associative algebras. For example, the operad of Novikov algebras is the derived one for the operad of (associative) commutative algebras. We state a sufficient condition for every algebra from a derived variety to be embeddable into an appropriate differential algebra of the corresponding variety. We also find that for ${\rm Var} = {\rm Zinb}$, the variety of Zinbiel algebras, there exist algebras from the derived variety (which coincides with the class of pre-Novikov algebras) that cannot be embedded into a Zinbiel algebra with a derivation.

Automorphisms of the universal enveloping algebra of a finite-dimensional Zinbiel algebra with zero multiplication

In recent years there has been a great interest in the study of Zinbiel (dual Leibniz) algebras. Let A be Zinbiel algebra over an arbitrary field K and let e1,e2,...,em,... be a linear basis of A. In 2010 A. Naurazbekova, using the methods of Gro¨bner-Shirshov bases, constructed the basis of the universal (multiplicative) enveloping algebra U(A) of A. Using this result, the automorphisms of the universal enveloping algebra of a finite-dimensional Zinbiel algebra with zero multiplication are described.

Local and 2-local derivations on small dimensional Zinbiel algebras

Local and 2-local derivations on small dimensional Zinbiel algebras

Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras

We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of O-operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of coherent derivations on an ASI bialgebra composes a Lie algebra which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.

The algebraic and geometric classification of Zinbiel algebras

This paper is devoted to the complete algebraic and geometric classification of complex 5-dimensional Zinbiel algebras. In particular, we proved that the variety of complex 5-dimensional Zinbiel algebras has dimension 24, it is defined by 16 irreducible components and it has 11 rigid algebras.

Abelian subalgebras and ideals of maximal dimension in Zinbiel algebras

In this paper, we compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional Zinbiel algebras. We study Zinbiel algebras containing maximal abelian subalgebras of codimension 1 and supersolvable Zinbiel algebras in which such subalgebras have codimension 2, and we also analyze the case of filiform Zinbiel algebras. We give examples to clarify some results, including listing the values for α and β for the low dimensional Zinbiel algebras over the complex field that have been classified. Communicated by Alberto Elduque

Algebras of slowly growing length

We investigate the class of finite-dimensional not necessarily associative algebras that have slowly growing length, that is, for any algebra in this class its length is less than or equal to its dimension. We show that this class is considerably big, in particular, finite-dimensional Lie algebras as well as many other important classical finite-dimensional algebras belong to this class, for example, Leibniz algebras, Novikov algebras and Zinbiel algebras. The exact upper bounds for the length of these algebras is proved. To do this, we transfer the method of characteristic sequences to non-unital algebras and find certain polynomial conditions on the algebra elements that guarantee the slow growth of the length function.

Symmetric Zinbiel superalgebras

The notion of symmetric Zinbiel superalgebras is introduced. We prove that the nilpotency index of a symmetric Zinbiel superalgebra is not greater than 4 and describe two-generated symmetric Zinbiel algebras and odd generated superalgebras. We discuss identities of mono and binary symmetric Zinbiel and Leibniz algebras. It is proven that each quadratic Zinbiel algebra is 2-step nilpotent. Also, we study double extensions of symmetric Zinbiel algebras.

Defining identities for mono and binary Zinbiel algebras

An algebra is said to be a mono Zinbiel algebra if each of its one-generated subalgebra is a Zinbiel algebra. An algebra is said to be a binary Zinbiel algebra if each of its two-generated subalgebra is a Zinbiel algebra. We give an independent set of defining identities for mono and binary Zinbiel algebras.

Zinbiel algebras are nilpotent

In this paper, we show that every finite-dimensional Zinbiel algebra over an arbitrary field is nilpotent, extending a previous result that they are solvable by other authors.

Zinbiel algebras and multiple zeta values

We build, using the notion of zinbiel algebra, some commutative subalgebras C_{u,v} inside an algebra of formal iterated integrals. There is a quotient map from this algebra of formal iterated integrals to the algebra of motivic multiple zeta values. Restricting this quotient map to the subalgebras C_{u,v} gives a morphism of graded commutative algebras with the same graded dimension. This is conjectured to be generically an isomorphism. When u+v = 0, the image is instead a sub-algebra of the algebra of motivic multiple zeta values.

Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebras

The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with multiple decorations. We also construct free commutative matching Rota-Baxter algebras in a relative context, to emulate the action of the integral operators on the coefficient functions in an integral equation. We finally show that free commutative matching Rota-Baxter algebras give the free matching Zinbiel algebra, generalizing the characterization of the shuffle product algebra as the free Zinbiel algebra obtained by Loday.

Central extensions of 3-dimensional Zinbiel algebras

We describe all central extensions of all $3$-dimensional non-zero complex Zinbiel algebras. As a corollary, we have a full classification of $4$-dimensional non-trivial complex Zinbiel algebras and a full classification of $5$-dimensional non-trivial complex Zinbiel algebras with $2$-dimensional annihilator, which gives the principal step in the algebraic classification of $5$-dimensional Zinbiel algebras.

Central extensions of filiform Zinbiel algebras

ABSTRACT In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an n-dimensional null-filiform Zinbiel algebra is isomorphic to an -dimensional null-filiform Zinbiel algebra. Moreover, we obtain all pairwise non isomorphic quasi-filiform Zinbiel algebras.

Signatures of paths transformed by polynomial maps

We characterize the signature of piecewise continuously differentiable paths transformed by a polynomial map in terms of the signature of the original path. For this aim, we define recursively an algebra homomorphism between two shuffle algebras on words. This homomorphism does not depend on the path and behaves well with respect to composition and homogeneous maps. We also study this map as a half-shuffle homomorphism and give a generalization of our main theorem in terms of Zinbiel algebras.

Non-associative structures of commutative algebras related with quadratic Poisson brackets

There are studied algebraic properties of quadratic Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Their relations both with derivations of symmetric tensor algebras and Yang–Baxter structures on the adjacent Lie algebras are demonstrated. Special attention is paid to quadratic Poisson brackets of Lie–Poisson type, examples of Balinsky–Novikov and Leibniz algebras are discussed. The non-associative structures of commutative algebras related with Balinsky–Novikov, Leibniz, Lie, and Zinbiel algebras are studied in detail.

Application of annihilator extension's method to classify Zinbiel algebras

Application of annihilator extension's method to classify Zinbiel algebras

Classification of Naturally Graded Zinbiel Algebras with Characteristic Sequence Equal to (n–p, p)

The present work continues our description of some classes of nilpotent Zinbiel algebras. We focus our attention on the investigation of Zinbiel algebras with restrictions imposed on the gradation and characteristic sequence. Namely, we propose a classification of naturally graded Zinbiel algebras with characteristic sequence equal to (n − p, p).