Weak composition quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras

Abstract

We introduce the Hopf algebra of quasi-symmetric functions with semigroup exponents generalizing the Hopf algebra QSym of quasi-symmetric functions. As a special case we obtain the Hopf algebra WCQSym of weak composition quasi-symmetric functions, which provides a framework for the study of a question proposed by G.-C.~Rota relating symmetric type functions and Rota-Baxter algebras. We provide the transformation formulas between the weak composition monomial and fundamental quasi-symmetric functions, which extends the corresponding results for quasi-symmetric functions. Moreover, we show that QSym is a Hopf subalgebra and a Hopf quotient algebra of WCQSym. Rota's question is addressed by identifying WCQsym with the free commutative unitary Rota-Baxter algebra of weight 1 on one generator, which also allows us to equip this algebra with a Hopf algebra structure.

25 pages

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• The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
• Rota-Baxter operators on Turaev's Hopf group (co)algebras I: Basic definitions and related algebraic structures
• On weak peak quasisymmetric functions
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• On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra
• On the product of generating functions for domino and bi-tableaux