New identities in dendriform algebras

Abstract

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 dialgebras 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.

16 pages, LaTeX. Concrete examples and applications added

References in corpus (7)

• Loday-type Algebras and the Rota-Baxter Relation
• A Lie theoretic approach to renormalization
• A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion
• Polynomial realizations of some trialgebras
• Infinitesimal bialgebras, pre-Lie and dendriform algebras
• Rota-Baxter algebras and new combinatorial identities
• Trees, functional equations, and combinatorial Hopf algebras

Cited by in corpus (20)

• Double constructions of Frobenius algebras, Connes cocycles and their duality
• A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion
• Some results on L-dendriform algebras
• A Magnus- and Fer-type formula in dendriform algebras
• Dendriform Equations
• Cumulants, free cumulants and half-shuffles
• Nonsymmetric operads in combinatorics
• Pluriassociative algebras II: The polydendriform operad and related operads
• Shuffle-compatible permutation statistics II: the exterior peak set
• BiHom-pre-alternative algebras and BiHom-alternative quadri-algebras
• The combinatorics of Bogoliubov's recursion in renormalization
• Cohomology and deformations of dendriform algebras, and ${\mathrm{Dend}}_{\infty }$-algebras
• Rota-Baxter Operators on pre-Lie superalgebras and beyond
• Cohomology and deformations of dendriform coalgebras
• Pluriassociative and polydendriform algebras
• Rota-Baxter Hom-Lie-admissible algebras
• Dynkin operators and renormalization group actions in pQFT
• Shuffle quadri-algebra and concatenation
• A one-parameter family of dendriform identities
• Double constructions of Heisenberg Frobenius algebras and Connes cocycles, and solutions of the three-dimensional associative Yang-Baxter equation