Dendriform Algebras Research Articles

Extending structures for dendriform algebras

In this paper, we devote to extending structures for dendriform algebras. First, we define extending datums and unified products of dendriform algebras, and theoretically solve the extending structure problem. As an application, we consider flag datums as a special case of extending structures, and give an example of the extending structure problem. Second, we introduce matched pairs and bicrossed products of dendriform algebras and theoretically solve the factorization problem for dendriform algebras. Moreover, we also introduce cocycle semidirect products and nonabelian semidirect products as special cases of unified products. Finally, we define the deformation map on a dendriform extending structure (more general case), not necessary a matched pair, which is more practical in the classifying complements problem.

Weighted differential (q-tri)dendriform algebras

In this paper, we first introduce a weighted derivation on algebras over an operad [Formula: see text], and prove that for the free [Formula: see text]-algebra, its weighted derivation is determined by the restriction on the generators. As applications, we propose the concept of weighted differential ([Formula: see text]-tri)dendriform algebras and study some basic properties of them. Then Novikov-(tri)dendriform algebras are initiated, which can be induced from differential ([Formula: see text]-tri)dendriform of weight zero. Finally, the corresponding free objects are constructed, in both the commutative and noncommutative contexts.

Matching relative Rota-Baxter algebras, matching dendriform algebras and their cohomologies

The notion of matching Rota-Baxter algebras was recently introduced by Gao, Guo, and Zhang motivated by the study of algebraic renormalization of regularity structures. In this paper, we introduce matching relative Rota-Baxter algebras that are closely related to matching dendriform algebras. We define the cohomology of a matching relative Rota-Baxter algebra using the classical Hochschild cohomology and a new cohomology induced by the matching operators. As an application, we show that our cohomology governs the formal deformation theory of the matching relative Rota-Baxter algebra. Finally, we define the cohomology of a matching dendriform algebra and study homotopy matching dendriform algebras.

L ∞ -structures and cohomology theory of compatible O-operators and compatible dendriform algebras

The notion of O-operator is a generalization of the Rota–Baxter operator in the presence of a bimodule over an associative algebra. A compatible O-operator is a pair consisting of two O-operators satisfying a compatibility relation. A compatible O-operator algebra is an algebra together with a bimodule and a compatible O-operator. In this paper, we construct a graded Lie algebra and an L∞-algebra that respectively characterize compatible O-operators and compatible O-operator algebras as Maurer–Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible O-operators and compatible O-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible O-operators.

Equivariant cohomology and deformation for algebras - an operadic approach

Let P be an operad with a multiplication, and later with a (higher) derivation. This paper introduces and develops a notion of (partial) group action and corresponding equivariant cohomology and deformation theory on P. It generalizes existing results for equivariant cohomology and deformation for associative algebras, dialgebras, and dendriform algebras, among others. At the end, we discuss compatibilities of group action and some operadic constructions.

Matching Rota–Baxter modules and matching 𝒪-operators of associative algebras

In this paper, firstly, we introduce the notion of matching Rota–Baxter modules, which generalizes the notions of matching Rota–Baxter algebras and Rota–Baxter modules, and give its characteristic and construction. Secondly, we give the relation between matching Rota–Baxter modules and other modules structure. Finally, we introduce the notion of matching [Formula: see text]-operators of associative algebras which also generalizes the notion of matching Rota–Baxter algebras and give the solution of polarized associative Yang–Baxter equation by using the matching [Formula: see text]-operators of associative algebras. In addition, we introduce the definitions of the compatible matching [Formula: see text]-operators of associative algebras, matching Nijenhui operators, and compatible matching dendriform algebras, and consider their connection.

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.

Hom-Leibniz bialgebras and BiHom-Leibniz dendriform algebras

A notion of a Hom-Leibniz bialgebra is introduced and it is shown that matched pairs of Hom-Leibniz algebras, Manin triples of Hom-Leibniz algebras and Hom-Leibniz bialgebras are equivalent in a certain sense. A notion of Hom-Leibniz dendriform algebra is established, their bimodules and matched pairs are defined and their properties and theorems about their interplay and construction are obtained. Furthermore, the concept of BiHom-Leibniz dendriform algebras is introduced and discussed, their bimodules and matched pairs are constructed and properties are described. Finally, the connections between all these algebraic structures using {mathcal {O}}-operators are shown.

Bimodules over Relative Rota-Baxter Algebras and Cohomologies

A relative Rota-Baxter algebra is a generalization of a Rota-Baxter algebra. Relative Rota-Baxter algebras are closely related to dendriform algebras. In this paper, we introduce bimodules over a relative Rota-Baxter algebra that fits with the representations of dendriform algebras. We define the cohomology of a relative Rota-Baxter algebra with coefficients in a bimodule and then study abelian extenfsions of relative Rota-Baxter algebras in terms of the second cohomology group. Finally, we consider homotopy relative Rota-Baxter algebras and classify skeletal homotopy relative Rota-Baxter algebras in terms of the above-defined cohomology.

Polarization and deformations of generalized dendriform algebras

We generalize three results of M. Aguiar on Loday’s dendriform algebras to dendriform algebras associated with algebras satisfying any given set of relations. We adapt the concept of polarization to such algebras, and use it to generalize Aguiar’s results on deformations and filtrations of dendriform algebras. We introduce weak Rota–Baxter operators and use them to prove a generalization of another result by Aguiar, which provides an interpretation of the natural relation between infinitesimal bialgebras and pre-Lie algebras in terms of dendriform algebras.

Cohomology and deformations of twisted Rota–Baxter operators and NS-algebras

The aim of this paper is twofold. In the first part, we consider twisted Rota–Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue of twisted Poisson structures. We construct an $$L_\infty $$ -algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This leads to cohomology associated to a twisted Rota–Baxter operator. This cohomology can be seen as the Hochschild cohomology of a certain associative algebra with coefficients in a suitable bimodule. We study deformations of twisted Rota–Baxter operators by means of the above-defined cohomology. Application is given to Reynolds operators. In the second part, we consider NS-algebras of Leroux that are related to twisted Rota–Baxter operators in the same way dendriform algebras are related to Rota–Baxter operators. We define cohomology of NS-algebras using non-symmetric operads and study their deformations in terms of the cohomology.

The L∞-deformations of associative Rota–Baxter algebras and homotopy Rota–Baxter operators

A relative Rota–Baxter algebra is a triple (A, M, T) consisting of an algebra A, an A-bimodule M, and a relative Rota–Baxter operator T. Using Voronov’s derived bracket and a recent work of Lazarev, Sheng, and Tang, we construct an L∞[1]-algebra whose Maurer–Cartan elements are precisely relative Rota–Baxter algebras. By a standard twisting, we define a new L∞[1]-algebra that controls Maurer–Cartan deformations of a relative Rota–Baxter algebra (A, M, T). We introduce the cohomology of a relative Rota–Baxter algebra (A, M, T) and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of triangular skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota–Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras.

Inner derivations of finite dimensional dendriform algebras

Inner derivations of finite dimensional dendriform algebras

Matching BiHom-Rota-Baxter Algebras and Related Structures

In this paper, we introduce the notions of matching BiHom-Rota-Baxter algebras, matching BiHom-(tri)dendriform algebras, matching BiHom-Zinbiel algebras and matching BiHom-pre-Lie algebras. Moreover, we study the properties and relationships between categories of these matching BiHom-algebraic structures.

Rota–Baxter operators on involutive associative algebras

In this paper, we consider Rota-Baxter operators on involutive associative algebras. We define cohomology for Rota-Baxter operators on involutive algebras that governs the formal deformation of the operator. This cohomology can be seen as the Hochschild cohomology of a certain involutive associative algebra with coefficients in a suitable involutive bimodule. We also relate this cohomology with the cohomology of involutive dendriform algebras. Finally, we show that the standard Fard-Guo construction of the functor from the category of dendriform algebras to the category of Rota-Baxter algebras restricts to the involutive case.

Cohomology and deformations of dendriform algebras, and Dend∞-algebras

A dendriform algebra is an associative algebra whose product splits into two binary operations and the associativity splits into three new identities. These algebras arise naturally from some combinatorial objects and through Rota-Baxter operators. In this paper, we start by defining explicit cohomology of dendriform algebras with coefficients in a representation. Our method avoids the heavy use of operad theory. Deformations of a dendriform algebra A are governed by the cohomology of A with coefficient in itself. Next, we study -algebras (dendriform algebras up to homotopy) in which the dendriform identities hold up to certain homotopy. They are a certain splitting of -algebras. We define Rota-Baxter operator on -algebras which naturally gives rise to -algebras. Finally, we classify skeletal and strict -algebras.

Generalizing dendriform algebras: Dyckm-algebras, Rotam-algebras, and Rota–Baxter operators

We review the notion of a [Formula: see text], an algebraic structure introduced recently by López, Préville-Ratelle and Ronco during their work on the splitting of associativity via [Formula: see text]-Dyck paths, and we also introduce Rota[Formula: see text]-algebras: both structures can be considered as generalizations of dendriform structures. We obtain examples of Dyck[Formula: see text]-algebras in terms of planar rooted binary trees equipped with a particular type of Rota–Baxter operator, and we present examples of Rotam-algebras using left averaging morphisms. As an application, we observe that the structures presented here allow us to introduce quite naturally a “non-associative version” of the Kadomtsev–Petviashvili hierarchy.

Typed binary trees and generalized dendrifom algebras

We here both unify and generalize nonassociative structures on typed binary trees, that is to say plane binary trees which edges are decorated by elements of a set Ω. We prove that we obtain such a structure, called an Ω-dendriform structure, if Ω has four products satisfying certain axioms (EDS axioms), including the axioms of a diassociative semigroup. This includes matching dendriform algebras introduced by Zhang, Gao and Guo and family dendriform algebras associated to a semigroup introduced by Zhang, Gao and Manchon, and of course dendriform algebras when Ω is reduced to a single element. We also give examples of EDS, including all the EDS of cardinality two; a combinatorial description of the products of such a structure on typed binary trees, but also on words; a study of the Koszul dual of the associated operads; and considerations on the existence of a coproduct, in order to obtain dendriform bialgebras.

Nonabelian extensions and factor systems for the algebras of Loday

Factor systems are a tool for working on the extension problem of algebraic structures such as groups, Lie algebras, and associative algebras. Their applications are numerous and well-known in these common settings. We construct algebra analogues to a series of results from W. R. Scott’s Group Theory, which gives an explicit theory of factor systems for the group case. Here ranges over Leibniz, Zinbiel, diassociative, and dendriform algebras, which we dub “the algebras of Loday,” as well as over Lie, associative, and commutative algebras. Fixing a pair of algebras, we develop a correspondence between factor systems and extensions. This correspondence is strengthened by the fact that equivalence classes of factor systems correspond to those of extensions. Under this correspondence, central extensions give rise to 2-cocycles while split extensions give rise to (nonabelian) 2-coboundaries.

Involutive and oriented dendriform algebras

Dendriform algebras are certain splitting of associative algebras and arise naturally from Rota-Baxter operators, shuffle algebras and planar binary trees. In this paper, we first consider involutive dendriform algebras, their cohomology and homotopy analogs. The cohomology of an involutive dendriform algebra splits the Hochschild cohomology of an involutive associative algebra. In the next, we introduce a more general notion of oriented dendriform algebras. We develop a cohomology theory for oriented dendriform algebras that closely related to extensions and governs the simultaneous deformations of dendriform structures and the orientation.