Rota-Baxter Algebras and Dendriform Algebras

Abstract

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.

arXiv commentTypos corrected and the last section on analog of Poincare-Birkhoff-Witt theorem deleted for a gap in the proof

References in corpus (22)

• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• Loday-type Algebras and the Rota-Baxter Relation
• Integrable Renormalization II: the general case
• Mixable Shuffles, Quasi-shuffles and Hopf Algebras
• Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
• A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion
• Free Rota-Baxter algebras and rooted trees
• Hopf algebra approach to Feynman diagram calculations
• New identities in dendriform algebras
• Rota-Baxter Algebras in Renormalization of Perturbative Quantum Field Theory
• Shuffle relations for regularised integrals of symbols
• Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion
• On Products and Duality of Binary, Quadratic, Regular Operads
• Combinatorics of renormalization as matrix calculus
• On some remarkable operads constructed from Baxter operators
• Renormalization of multiple zeta values
• Matrix Representation of Renormalization in Perturbative Quantum Field Theory
• Renormalization of Multiple $q$-Zeta Values
• Completing the operadic butterfly
• On matrix differential equations in the Hopf algebra of renormalization
• Coherent Unit Actions on Operads and Hopf Algebras

Cited by in corpus (57)

• Splitting of operations, Manin products and Rota-Baxter operators
• Mixable Shuffles, Quasi-shuffles and Hopf Algebras
• Free Rota-Baxter algebras and rooted trees
• On embedding of dendriform algebras into Rota---Baxter algebras
• Differential Type Operators and Gröbner-Shirshov Bases
• Quillen homology for operads via Gröbner bases
• The algebra of integro-differential operators on a polynomial algebra
• Averaging algebras, Schröder numbers, rooted trees and operads
• Twisted dendriform algebras and the pre-Lie Magnus expansion
• Rota-Baxter Operators on Quadratic Algebras
• Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras
• Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases
• Universal enveloping Rota-Baxter algebras of preassociative and postassociative algebras
• Generalized shuffles related to Nijenhuis and TD-algebras
• Free resolutions via Gröbner bases
• Construction of Rota-Baxter algebras via Hopf module algebras
• Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gröbner-Shirshov bases
• Rota-Baxter Coalgebras
• Composition-Diamond lemma for $\lambda$-differential associative algebras with multiple operators
• Differential Birkhoff decomposition and the renormalization of multiple zeta values
• Polylogarithms and multiple zeta values from free Rota-Baxter algebras
• A Model of Cooperative Threads
• Rota-Baxter operators on BiHom-associative algebras and related structures
• Construction of free differential algebras by extending Gröbner-Shirshov bases
• Representations of Rota-Baxter algebras and regular singular decompositions
• Transposed Hom-Poisson and Hom-pre-Lie Poisson algebras and bialgebras
• Free objects and Gröbner-Shirshov bases in operated contexts
• Cohomology of weighted Rota-Baxter Lie algebras and Rota-Baxter paired operators
• Cohomology and deformations of dendriform algebras, and ${\mathrm{Dend}}_{\infty }$-algebras
• Splitting of operads and Rota-Baxter operators on operads
• Free Rota-Baxter systems and a Hopf algebra structure
• Free Rota-Baxter family algebras and Free (tri)dendriform family algebras
• O-operators on associative algebras and dendriform algebras
• On CSP and the Algebraic Theory of Effects
• Rota-Baxter Operators on pre-Lie superalgebras and beyond
• Structure theorems of mixable shuffle algebras and free commutative Rota-Baxter algebras
• Bialgebra and Hopf algebra structures on free Rota-Baxter algebras
• Braided Rota-Baxter algebras, quantum quasi-shuffle algebras and braided dendriform algebras
• Left counital Hopf algebra structures on free commutative Nijenhuis algebras
• Quasi-idempotent Rota-Baxter operators arising from quasi-idempotent elements
• Embedding dendriform dialgebra into its universal enveloping Rota-Baxter algebra
• Gröbner-Shirshov bases for Rota-Baxter algebras
• Gröbner-Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras (unabridged version)
• Hopf algebras of rooted forests, cocyles and free Rota-Baxter algebras
• The global dimension of the algebras of polynomial integro-differential operators ${\mathbb{I}}_n$ and the Jacobian algebras ${\mathbb{A}}_n$
• Modules of polynomial Rota-Baxter Algebras and matrix equations
• Reynolds algebras and their free objects from bracketed words and rooted trees
• Cohomology and deformations of dendriform coalgebras
• Embedding of post-Lie algebras into postassociative algebras
• Characteristics of Rota-Baxter Algebras
• Splitting of operations for alternative and Malcev structures
• Rota-Baxter Hom-Lie-admissible algebras
• Embedding of pre-Lie algebras into preassociative algebras
• Rota-Baxter operators on generalized power series rings
• Derived identities of differential algebras
• Commutative modified Rota-Baxter algebras, shuffle products and Hopf algebras
• Nijenhuis algebras, NS algebras and N-dendriform algebras