Dendriform Algebras Research Articles

Rota–Baxter Hom-Lie–Admissible Algebras

We study Hom-type analogs of Rota–Baxter and dendriform algebras, called Rota–Baxter G-Hom–associative algebras and Hom-dendriform algebras. Several construction results are proved. Free algebras for these objects are explicitly constructed. Various functors between these categories, as well as an adjunction between the categories of Rota–Baxter Hom-associative algebras and of Hom-(tri)dendriform algebras, are constructed.

Gröbner–Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras

In this paper we use the operadic framework to find Gröbner–Shirshov bases for the free quadri-algebra. We perform computations using the representation of the non-symmetric operad by planar rooted trees in a very intuitive way. Gröbner–Shirshov bases for the free dendriform algebra are also found with this technique, simplifying the work by Chen and Wang (2010).

Pre-jordan Algebras

The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegenerate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of $\mathcal{O}$-operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a pre-Jordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of $\mathcal{O}$-operator of a pre-Jordan algebra which gives an analogue of the classical Yang-Baxter equation in a pre-Jordan algebra.

Embedding of dendriform algebras into Rota-Baxter algebras

Abstract Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.

Twisted dendriform algebras and the pre-Lie Magnus expansion

In this paper an application of the recently introduced pre-Lie Magnus expansion to Jackson’s q -integral and q -exponentials is presented. Twisted dendriform algebras, which are the natural algebraic framework for Jackson’s q -analogs, are introduced for that purpose. It is shown how the pre-Lie Magnus expansion is used to solve linear q -differential equations. We also briefly outline the theory of linear equations in twisted dendriform algebras.

𝒪-Operators of Loday Algebras and Analogues of the Classical Yang–Baxter Equation

We introduce notions of 𝒪-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota–Baxter operators. The invertible 𝒪-operators give a sufficient and necessary condition on the existence of the 2 n+1 operations on an algebra with the 2 n operations in an associative cluster. The analogues of the classical Yang–Baxter equation in these algebras can be understood as the 𝒪-operators associated to certain dual bimodules. As a byproduct, the constraint conditions (invariances) of nondegenerate bilinear forms on these algebras are given.

Double constructions of Frobenius algebras, Connes cocycles and their duality

We construct an associative algebra with a decomposition into the direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is a Connes cocycle. The former is called a double construction of a Frobenius algebra and the latter is called a double construction of the Connes cocycle, which is interpreted in terms of dendriform algebras. Both of them are equivalent to a kind of bialgebras, namely, antisymmetric infinitesimal bialgebras and dendriform D-bialgebras, respectively. In the coboundary cases, our study leads to what we call associative Yang–Baxter equation in an associative algebra and D -equation in a dendriform algebra, respectively, which are analogues of the classical Yang–Baxter equation in a Lie algebra. We show that an antisymmetric solution of the associative Yang–Baxter equation corresponds to the antisymmetric part of a certain operator called \mathcal O -operator which gives a double construction of a Frobenius algebra, whereas a symmetric solution of the D -equation corresponds to the symmetric part of an \mathcal O -operator which gives a double construction of the Connes cocycle. By comparing antisymmetric infinitesimal bialgebras and dendriform D-bialgebras, we observe that there is a clear analogy between them. Due to the correspondences between certain symmetries and antisymmetries appearing in this analogy, we regard it as a kind of duality.

Some results on L-dendriform algebras

We introduce the notion of an L-dendriform algebra due to several different motivations. L-dendriform algebras are regarded as the underlying algebraic structures of pseudo-Hessian structures on Lie groups and the algebraic structures behind the O -operators of pre-Lie algebras and the related S -equation. As a direct consequence, they provide some explicit solutions of S -equations in certain pre-Lie algebras constructed from L-dendriform algebras. They also fit into a bigger framework as Lie algebraic analogues of dendriform algebras. Moreover, we introduce the notion of an O -operator of an L-dendriform algebra which gives an algebraic equation regarded as an analogue of the classical Yang–Baxter equation in a Lie algebra.

Generalized Shuffles Related to Nijenhuis and TD-Algebras

Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free Rota–Baxter algebras. In the literature there exist at least two other Rota–Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators. Our construction is related to Loday's unital commutative tridendriform algebra, including the involutive case. The concept of Rota–Baxter, Nijenhuis and TD-bialgebras is introduced at the end, and we show that any commutative bialgebra provides such objects.

Dendriform equations

We investigate solutions for a particular class of linear equations in dendriform algebras. Motivations as well as several applications are provided. The latter follow naturally from the intimate link between dendriform algebras and Rota–Baxter operators, e.g. the Riemann integral map or Jackson's q-integral.

Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota–Baxter Operators

We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a “generalized” Rota–Baxter identity of weight zero. Such operators are called generalized Rota–Baxter operators. We will show that generalized Rota–Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator “twisted Rota–Baxter operators,” which is a natural generalization of generalized Rota–Baxter operators. It is known that classical Rota–Baxter operators are closely related with dendriform algebras. We will show that twisted Rota–Baxter operators induce NS-algebra, which is a twisted version of dendriform algebra. The twisted Poisson condition is considered as a Maurer–Cartan equation up to homotopy. We will show the twisted Rota–Baxter condition also is so. And we will study a Poisson-geometric reason, how the twisted Rota–Baxter condition arises.

A Magnus- and Fer-Type Formula in Dendriform Algebras

We provide a refined approach to the classical Magnus (Commun. Pure Appl. Math. 7:649–673, [1954]) and Fer expansion (Bull. Classe Sci. Acad. R. Belg. 44:818–829, [1958]), unveiling a new structure by using the language of dendriform and pre-Lie algebras. The recursive formula for the logarithm of the solutions of the equations X=1+λ a ≺ X and Y=1−λ Y ≻ a in A[[λ]] is provided, where (A,≺,≻) is a dendriform algebra. Then we present the solutions to these equations as an infinite product expansion of exponentials. Both formulae involve the pre-Lie product naturally associated with the dendriform structure. Several applications are presented.

New identities in dendriform algebras

Dendriform structures arise naturally in algebraic combinatorics (where they allow, for example, the splitting of the shuffle product into two pieces) and through Rota–Baxter algebra structures (the latter appear, among others, in differential systems and in the renormalization process of pQFT). We prove new combinatorial identities in dendriform algebras that appear to be strongly related to classical phenomena, such as the combinatorics of Lyndon words, rewriting rules in Lie algebras, or the fine structure of the Malvenuto–Reutenauer algebra. One of these identities is an abstract noncommutative, dendriform, generalization of the Bohnenblust–Spitzer identity and of an identity involving iterated Chen integrals due to C.S. Lam.

Rota–Baxter algebras and dendriform algebras

In this paper we study the adjoint functors between the category of Rota–Baxter algebras and the categories of dendriform dialgebras and trialgebras. In analogy to the well-known theory of the adjoint functor between the category of associative algebras and Lie algebras, we first give an explicit construction of free Rota–Baxter algebras and then apply it to obtain universal enveloping Rota–Baxter algebras of dendriform dialgebras and trialgebras. We further show that free dendriform dialgebras and trialgebras, as represented by binary planar trees and planar trees, are canonical subalgebras of free Rota–Baxter algebras.

Ennea-algebras

A generalisation of a recent work of M. Aguiar and J.-L. Loday on quadrialgebras called t-ennea-algebras constructed over dendriform trialgebras is proposed. Such algebras allow the construction of nested dendriform trialgebras and are related to pre-Lie algebras, t-infinitesimal bialgebras and t-Baxter operators. We also show that the augmented free t-ennea-algebra has a structure of connected Hopf algebra. In the last part, we use Baxter operators to produce formal deformations of dendriform algebras, quadrialgebras and ennea-algebras. Examples and relations of this work to combinatorial objects are given.

Quadri-algebras

We introduce the notion of quadri-algebras. These are associative algebras for which the multiplication can be decomposed as the sum of four operations in a certain coherent manner. We present several examples of quadri-algebras: the algebra of permutations, the shuffle algebra, tensor products of dendriform algebras. We show that a pair of commuting Baxter operators on an associative algebra gives rise to a canonical quadri-algebra structure on the underlying space of the algebra. The main example is provided by the algebra End(A) of linear endomorphisms of an infinitesimal bialgebra A. This algebra carries a canonical pair of commuting Baxter operators: β(T)=T∗ id and γ(T)= id∗T , where ∗ denotes the convolution of endomorphisms. It follows that End(A) is a quadri-algebra, whenever A is an infinitesimal bialgebra. We also discuss commutative quadri-algebras and state some conjectures on the free quadri-algebra.

Eulerian idempotents and Milnor–Moore theorem for certain non-cocommutative Hopf algebras

The purpose of this paper is to prove a Milnor–Moore style theorem for a particular kind of non-cocommutative Hopf algebras: the dendriform algebras. A dendriform Hopf algebra is a Hopf algebra, such that the product ∗ is the sum of two operations ≺ and ≻, verifying certain conditions between them and with the coproduct Δ. The role of Lie algebras is played by brace algebras, which are defined by n-ary operations (one for each n⩾2) satisfying some relations. We show that a dendriform Hopf algebra is isomorphic to the enveloping algebra of its brace algebra of primitive elements. One of the ingredients of the proof is the construction of Eulerian idempotents in this context.