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

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 QSymN˜ of weak quasi-symmetric functions, which provides a framework for the study of a question proposed...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2019-02, Vol.344, p.1-34
Main Authors: Yu, Houyi, Guo, Li, Thibon, Jean-Yves
Format: Article
Language:English
Subjects:
Hopf algebras
Mathematics
Quasi-symmetric functions
Rota–Baxter algebras
Symmetric functions
Weak compositions
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Summary: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 QSymN˜ of weak 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 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 QSymN˜. Rota's question is addressed by identifying QSymN˜ with the free commutative unitary Rota–Baxter algebra Ш(x) of weight 1 on one generator x, which also allows us to equip Ш(x) with a Hopf algebra structure.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2018.12.001