On embedding of dendriform algebras into Rota---Baxter algebras

Abstract

Following a recent work by C. Bai, O. Bellier, L. Guo, and X. Ni (arXiv:1106.6080) we define what is a dendriform di- or trialgebra in an arbitrary variety Var of algebras (associative, commutative, Poisson, etc.). We prove that every dendriform dialgebra in Var can be embedded into a Rota---Baxter algebra of weight zero in the same variety, and every dendriform trialgebra can be embedded into a Rota---Baxter algebra of nonzero weight.

v3: a few new proofs added

References in corpus (8)

• Loday-type Algebras and the Rota-Baxter Relation
• Splitting of operations, Manin products and Rota-Baxter operators
• Mixable Shuffles, Quasi-shuffles and Hopf Algebras
• Some results on L-dendriform algebras
• On Products and Duality of Binary, Quadratic, Regular Operads
• Derived bracket construction and Manin products
• O-operators on associative algebras and dendriform algebras
• Gröbner-Shirshov bases for Rota-Baxter algebras

Cited by in corpus (30)

• Rota-Baxter groups, skew left braces, and the Yang-Baxter equation
• Averaging algebras, Schröder numbers, rooted trees and operads
• Rota-Baxter operators on groups
• Rota-Baxter Operators on Quadratic Algebras
• Rota-Baxter operators on unital algebras
• Dendriform Algebras Relative to a Semigroup
• Universal enveloping Rota-Baxter algebras of preassociative and postassociative algebras
• Groebner-Shirshov bases for pre-associative algebras
• Homogeneous conformal averaging operators on semisimple Lie algebras
• Groebner---Shirshov bases for replicated algebras
• BiHom-pre-alternative algebras and BiHom-alternative quadri-algebras
• Replicating of binary operads, Koszul duality, Manin products and average operators
• Spectrum of Rota-Baxter operators
• Cohomology and deformations of dendriform algebras, and ${\mathrm{Dend}}_{\infty }$-algebras
• Operads of decorated trees and their duals
• Algebras, dialgebras, and polynomial identities
• Groebner-Shirshov basis of the universal enveloping Rota-Baxter algebra of a Lie algebra
• Universal enveloping Lie Rota-Baxter algebra of preLie and post-Lie algebras
• Dendriform analogues of Lie and Jordan triple systems
• On the computation of Manin products for operads
• Reynolds algebras and their free objects from bracketed words and rooted trees
• Embedding of post-Lie algebras into postassociative algebras
• Relative (pre-)anti-flexible algebras and associated algebraic structures
• On the Dong Property for a binary quadratic operad
• Derived identities of differential algebras
• Rota-Baxter operators on $\mathrm{Cur}({\mathrm{sl}}_2(\mathbb{C}))$
• Embedding of pre-Lie algebras into preassociative algebras
• Rota-Baxter operators and Bernoulli polynomials
• Rota-Baxter coalgebras and Rota-Baxter bialgebras
• Quadratic Leibniz conformal algebras