Dendriform Algebras Research Articles

Braided Rota–Baxter algebras, quantum quasi-shuffle algebras and braided dendriform algebras

Rota–Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota–Baxter algebras and dendriform algebras in the braided contexts. Applying the quantum shuffle and quantum quasi-shuffle products, we construct free objects in the categories of braided Rota–Baxter algebras and braided dendriform algebras, under the commutativity condition. We further generalize the notion of dendriform Hopf algebra to the braided context and show that quantum shuffle algebra gives a braided dendriform Hopf algebra. Enveloping braided commutative Rota–Baxter algebras of braided commutative dendriform algebras are obtained.

On equivariant dendriform algebras

Dendriform algebras are certain associative algebras whose product splits into two binary operations and the associativity splits into three new identities. In this paper, we study finite group actions on dendriform algebras. We define equivariant cohomology for dendriform algebras equipped with finite group actions similar to the Bredon cohomology for topological $G$-spaces. We show that equivariant cohomology of such dendriform algebras controls equivariant one-parameter formal deformations.

Cohomology and deformations of hom-dendriform algebras and coalgebras

Hom-dendriform algebras are twisted analog of dendriform algebras and are splitting of hom-associative algebras. In this paper, we define a cohomology and deformation for hom-dendriform algebras. We relate this cohomology with the Hochschild-type cohomology of hom-associative algebras. We also describe similar results for the twisted analog of dendriform coalgebras.

Dendriform Algebras Relative to a Semigroup

Loday's dendriform algebras and its siblings pre-Lie and zinbiel have received attention over the past two decades. In recent literature, there has been interest in a generalization of these types of algebra in which each individual operation is replaced by a family of operations indexed by a fixed semigroup $S$. The purpose of this note is twofold. First, we add to the existing work by showing that a similar extension is possible already for the most familiar types of algebra: commutative, associative, and Lie. Second, we show that these concepts arise naturally and in a unified manner from a categorical perspective. For this, one simply has to consider the standard types of algebra but in reference to the monoidal category of $S$-graded vector spaces.

Matching Hom-Setting of Rota-Baxter Algebras, Dendriform Algebras, and Pre-Lie Algebras

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

Free quadri-algebras and dual quadri-algebras

We study quadri-algebras and dual quadri-algebras. We describe the free quadri-algebra on one generator as a subobject of the Hopf algebra of permutations FQSym, proving a conjecture due to Aguiar and Loday, using that the operad of quadri-algebras can be obtained from the operad of dendriform algebras by both black and white Manin products. We also give a combinatorial description of free dual quadri-algebras. A notion of quadri-bialgebra is also introduced, with the applications to the Hopf algebras FQSym and WQSym.

Braided dendriform and tridendriform algebras, and braided Hopf algebras of rooted trees

This paper studies the braidings of several Hopf algebras of rooted trees which have found broad applications. First by studying free braided dendriform algebras, we obtain the braiding of the Loday–Ronco Hopf algebra of planar binary rooted trees. We also give a variation of the braiding obtained by Foissy for the noncommutative Connes–Kreimer (a.k.a the Foissy–Holtkamp) Hopf algebra of planar rooted forests, and show that the well-known isomorphism between this Hopf algebra and the Loday–Ronco Hopf algebra can be extended to the braided context. We further study braided tridendriform algebras and apply their free objects to give a braiding of the Hopf algebra of Loday and Ronco on planar angularly decorated trees. Along the way, we obtain a combinatorial characterization and a generating function for such trees.

Rota–Baxter cosystems and coquasitriangular mixed bialgebras

In this paper, we present a dual version of T. Brzeziński’s results about Rota–Baxter systems which appeared in [Rota–Baxter systems, dendriform algebras and covariant bialgebras, J. Algebra 460 (2016) 1–25]. Then as a generalization to bialgebras, we introduce the notion of Rota–Baxter bisystem and construct various examples of Rota–Baxter bialgebras and bisystems in dimensions 2, 3 and 4. On the other hand, we introduce a new type of bialgebras (named mixed bialgebras) which consist of an associative algebra and a coassociative coalgebra satisfying the compatible condition determined by two coderivations. We investigate coquasitriangular mixed bialgebras and the particular case of coquasitriangular infinitesimal bialgebras, where we give the double construction. Also, we show in some cases that Rota–Baxter cosystems can be obtained from a coquasitriangular mixed bialgebras.

Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras

We introduce the notion of a matching Rota-Baxter algebra motivated by the recent work on multiple pre-Lie algebras arising from the study of algebraic renormalization of regularity structures [10,19]. This notion is also related to iterated integrals with multiple kernels and solutions of the associative polarized Yang-Baxter equation. Generalizing the natural connection of Rota-Baxter algebras with dendriform algebras to matching Rota-Baxter algebras, we obtain the notion of matching dendriform algebras. As in the classical case of one operation, matching Rota-Baxter algebras and matching dendriform algebras are related to matching pre-Lie algebras which coincide with the aforementioned multiple pre-Lie algebras. More general notions and results on matching tridendriform algebras and matching PostLie algebras are also obtained.

Endofunctors and Poincaré–Birkhoff–Witt Theorems

AbstractWe determine what appears to be the bare-bones categorical framework for Poincaré–Birkhoff–Witt (PBW)-type theorems about universal enveloping algebras of various algebraic structures. Our language is that of endofunctors; we establish that a natural transformation of monads enjoys a PBW property only if that transformation makes its codomain a free right module over its domain. We conclude with a number of applications to show how this unified approach proves various old and new PBW-type theorems. In particular, we prove a PBW-type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.

Nijenhuis operators on pre-Lie algebras

First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.

A simplicial complex splitting associativity

We introduce a simplicial object ({Dyckm}m≥0,Fi,Sj) in the category of non-symmetric algebraic operads, satisfying that Dyck0 is the operad of associative algebras and Dyck1 is J.-L. Loday’s operad of dendriform algebras. The dimensions of the operad Dyckm are given by the Fuss-Catalan numbers.Given a family of partially ordered sets P={Pn}n≥1 we show that, under certain conditions, the vector space spanned by the set of m-simpleces of P is a Dyckm algebra. This construction, applied to certain combinatorial Hopf algebras, whose associative product comes from a dendriform structure, provides examples of Dyckm algebras.

On derivations of finite dimensional dendriform algebras

On derivations of finite dimensional dendriform algebras

Solutions of associative Yang–Baxter equation and D−equation in low dimensions and associated Frobenius algebras and Connes cocycles

In this work, we compute solutions of the Yang–Baxter associative equation in dimensions one and two. For these solutions, we describe the double constructions of the associated Frobenius algebras, following Bai’s definitions. Besides, we determine related compatible dendriform algebras and solutions of their [Formula: see text]equations. Finally, using symmetric solutions of these equations, we build the double constructions of related Connes cocycles.

Gröbner–Shirshov bases for pre-associative algebras

ABSTRACTWe develop Gröbner–Shirshov bases technique for pre-associative (dendriform) algebras and prove a version of composition-diamond lemma.

Rota–Baxter systems, dendriform algebras and covariant bialgebras

A generalisation of the notion of a Rota–Baxter operator is proposed. This generalisation consists of two operators acting on an associative algebra and satisfying equations similar to the Rota–Baxter equation. Rota–Baxter operators of any weights and twisted Rota–Baxter operators are solutions of the proposed system. It is shown that dendriform algebra structures of a particular kind are equivalent to Rota–Baxter systems. It is shown further that a Rota–Baxter system induces a weak pseudotwistor [Panaite and Van Oystaeyen (2015) [15]] which can be held responsible for the existence of a new associative product on the underlying algebra. Examples of solutions of Rota–Baxter systems are obtained from quasitriangular covariant bialgebras hereby introduced as a natural extension of infinitesimal bialgebras [Aguiar (2000) [3]].

Pluriassociative algebras II: The polydendriform operad and related operads

Dendriform algebras form a category of algebras recently introduced by Loday. A dendriform algebra is a vector space endowed with two nonassociative binary operations satisfying some relations. Any dendriform algebra is an algebra over the dendriform operad, the Koszul dual of the diassociative operad. We introduce here, by adopting the point of view and the tools offered by the theory of operads, a generalization on a nonnegative integer parameter γ of dendriform algebras, called γ-polydendriform algebras, so that 1-polydendriform algebras are dendriform algebras. For that, we consider the operads obtained as the Koszul duals of the γ-pluriassociative operads introduced by the author in a previous work. In the same manner as dendriform algebras are suitable devices to split associative operations into two parts, γ-polydendriform algebras seem adapted structures to split associative operations into 2γ operation so that some partial sums of these operations are associative. We provide a complete study of the γ-polydendriform operads, the underlying operads of the category of γ-polydendriform algebras. We exhibit several presentations by generators and relations, compute their Hilbert series, and construct free objects in the corresponding categories. We also provide consistent generalizations on a nonnegative integer parameter of the duplicial, triassociative and tridendriform operads, and of some operads of the operadic butterfly.

Splitting of Operads and Rota-Baxter Operators on Operads

This paper establishes a procedure that splits the operations in any algebraic operad, generalizing previous notions of splitting algebraic structures, from the dendriform algebra of Loday splitting the associative operation to the successors splitting binary operads. The separately treated bisuccessor and trisuccessor for binary operads are unified for general operads through the notion of configuration. Applications are provided for various n-algebras, the \(A_{\infty }\) and \(L_{\infty }\) algebras. Further, the concept of a Rota-Baxter operator, first showing its importance in the associative and Lie algebra contexts and then generalized to binary operads, is defined for all operads. The well-known connection from Rota-Baxter operators to dendriform algebras and its numerous extensions are expanded as the link from (relative) Rota-Baxter operators on operads to splittings of the operads

Combinatorial proofs of freeness of some P-algebras

We present new combinatorial methods for solving algebraic problems such as computing the Hilbert series of a free $P$-algebra over one generator, or proving the freeness of a $P$-algebra. In particular, we apply these methods to the cases of dendriform algebras, quadrialgebras and tridendriform algebras, which leads us to prove a conjecture of Aguiar and Loday about the freeness of the quadrialgebra generated by the permutation 12. Nous présentons de nouvelles méthodes combinatoires permettant de résoudre des problèmes algébriques concernant les $P$-algèbres, comme déterminer la série de Hilbert de la $P$-algèbre libre sur un générateur, ou de prouver qu’une $P$-algèbre est libre. Nous les appliquons aux cas des algèbres dendriformes, des quadrialgèbres, et des algèbres tridendriformes. Cette approche nous permet en particulier de résoudre une conjecture de Aguiar et Loday à propos de la liberté de la quadrialgèbre engendrée par la permutation 12.

Centroids of Zinbiel Algebras

Zinbiel algebras were introduced by a way of Koszul dual to Leibniz algebras and connected with commutative associate algebras, pre-Lie algebras and dendriform algebras etc. In this article, the centroids of Zinbiel algebra are discussed. We study the proprieties of centroids and also determinate the centroids of Zinbiel algebras up to dimension ≤4. As consequent, we get the conclusion that the centroids of 2-dimensional Zinbiel algebras are not small and 3,4-dimensional Zinbiel algebras are almost small.