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 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.

[1]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[2]  Kurusch Ebrahimi-Fard,et al.  A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion , 2007 .

[3]  Jean-Yves Thibon,et al.  Trees, functional equations, and combinatorial Hopf algebras , 2008, Eur. J. Comb..

[4]  Guy Melançon,et al.  Lyndon Words, Free Algebras and Shuffles , 1989, Canadian Journal of Mathematics.

[5]  Gian-Carlo Rota,et al.  Baxter algebras and combinatorial identities. II , 1969 .

[6]  F. Patras,et al.  A Lie Theoretic Approach to Renormalization , 2006, hep-th/0609035.

[7]  Maria O. Ronco.  Eulerian idempotents and Milnor–Moore theorem for certain non-cocommutative Hopf algebras , 2002 .

[8]  M. AGUIAR,et al.  Infinitesimal Hopf algebras , 2007 .

[9]  H. Schneider,et al.  New Trends in Hopf Algebra Theory , 2000 .

[10]  Maria O. Ronco,et al.  Primitive elements in a free dendriform algebra , 1999 .

[11]  Alain Lascoux,et al.  Noncommutative symmetric functions , 1994 .

[12]  K. Ebrahimi-Fard.  Loday-Type Algebras and the Rota–Baxter Relation , 2002 .

[13]  Christophe Reutenauer,et al.  On Dynkin and Klyachko Idempotents in Graded Bialgebras , 2002, Adv. Appl. Math..

[14]  Frederic Chapoton.  Un théorème de Cartier–Milnor–Moore–Quillen pour les bigèbres dendriformes et les algèbres braces , 2000 .

[15]  Israel M. Gelfand,et al.  Noncommutative Symmetrical Functions , 1995 .

[16]  J. Thibon,et al.  Polynomial realizations of some trialgebras , 2006, math/0605061.

[17]  John Milnor,et al.  On the Structure of Hopf Algebras , 1965 .

[18]  Marcel P. Schützenberger,et al.  Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathématiques appliquées , 1959 .

[19]  Marcelo Aguiar,et al.  Pre-Poisson Algebras , 2000 .

[20]  Tero Harju,et al.  Combinatorics on Words , 2004 .

[21]  F. Patras.  Decent Algebra of a Graduated Bigebra , 1994 .

[22]  Loic Foissy,et al.  Bidendriform bialgebras, trees, and free quasi-symmetric functions , 2005, math/0505207.

[23]  R. Ree,et al.  Lie Elements and an Algebra Associated With Shuffles , 1958 .

[24]  Jean-Louis Loday,et al.  Trialgebras and families of polytopes , 2002 .

[25]  C. Reutenauer,et al.  Duality between Quasi-Symmetrical Functions and the Solomon Descent Algebra , 1995 .

[26]  Marcelo Aguiar,et al.  On the Associative Analog of Lie Bialgebras , 2001 .

[27]  Frédéric Patras,et al.  Rota–Baxter Algebras and New Combinatorial Identities , 2006, math/0701031.