ALGEBRAIC COMBINATORICS

no. 4
Demi-shuffle duals of Magnus polynomials in a free associative algebra
Nakamura, Hiroaki1
1 Osaka University Department of Mathematics, Graduate School of Science Toyonaka, Osaka 560-0043 (Japan)
Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 929-939.

We study two linear bases of the free associative algebra ℤ〈X,Y〉: one is formed by the Magnus polynomials of type (ad X k 1 Y)⋯(ad X k d Y)X k and the other is its dual basis (formed by what we call the “demi-shuffle” polynomials) with respect to the standard pairing on the monomials of ℤ〈X,Y〉. As an application, we derive a formula of Le–Murakami, Furusho type that expresses arbitrary coefficients of a group-like series J∈ℂ〈〈X,Y〉〉 in terms of the “regular” coefficients of J.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.287
Classification: 10X99, 14A12, 11L05
Keywords: shuffle product, non-commutative polynomial, group-like series
Nakamura, Hiroaki 1

1 Osaka University Department of Mathematics, Graduate School of Science Toyonaka, Osaka 560-0043 (Japan)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Nakamura, Hiroaki. Demi-shuffle duals of Magnus polynomials in a free associative algebra. Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 929-939. doi : 10.5802/alco.287. https://alco.centre-mersenne.org/articles/10.5802/alco.287/

[1] Chapoton, Frédéric Zinbiel algebras and multiple zeta values, Doc. Math., Volume 27 (2022), pp. 519-533 | DOI | MR | Zbl

[2] Dokas, Ioannis Zinbiel algebras and commutative algebras with divided powers, Glasg. Math. J., Volume 52 (2010) no. 2, pp. 303-313 | DOI | MR | Zbl

[3] Foissy, Loïc; Patras, Frédéric Natural endomorphisms of shuffle algebras, Internat. J. Algebra Comput., Volume 23 (2013) no. 4, pp. 989-1009 | DOI | MR | Zbl

[4] Furusho, Hidekazu p-adic multiple zeta values. I. p-adic multiple polylogarithms and the p-adic KZ equation, Invent. Math., Volume 155 (2004) no. 2, pp. 253-286 | DOI | MR | Zbl

[5] Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren Concrete mathematics, Addison-Wesley Publishing Company, Reading, MA, 1994, xiv+657 pages (A foundation for computer science)

[6] Hoang Ngoc Minh, Vincel On the solutions of the universal differential equation with three regular singularities (on solutions of KZ 3 ), Confluentes Math., Volume 11 (2019) no. 2, pp. 25-64 | DOI | MR | Zbl

[7] Le, Thang Tu Quoc; Murakami, Jun Kontsevich’s integral for the Kauffman polynomial, Nagoya Math. J., Volume 142 (1996), pp. 39-65 | DOI | MR | Zbl

[8] Loday, Jean-Louis Cup-product for Leibniz cohomology and dual Leibniz algebras, Math. Scand., Volume 77 (1995) no. 2, pp. 189-196 | DOI | MR | Zbl

[9] Magnus, Wilhelm Über Beziehungen zwischen höheren Kommutatoren, J. Reine Angew. Math., Volume 177 (1937), pp. 105-115 | DOI | MR | Zbl

[10] Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald Combinatorial group theory, Dover Publications, Inc., Mineola, NY, 2004, xii+444 pages (Presentations of groups in terms of generators and relations)

[11] Maplesoft, a division of Waterloo Maple Inc.. Maple (2021 version) https://hadoop.apache.org

[12] Minh, Hoang Ngoc; Petitot, Michel; Van Der Hoeven, Joris Shuffle algebra and polylogarithms, Discrete Math., Volume 225 (2000) no. 1-3, pp. 217-230 Formal power series and algebraic combinatorics (Toronto, ON, 1998) | DOI | MR | Zbl

[13] Nakamura, Hiroaki Some aspects of arithmetic functions in Grothendieck-Teichmüller theory, Oberwolfach Rep., Volume 18 (2021) no. 1, pp. 700-702 part of “Homotopic and geometric Galois theory. Abstracts from the workshop held March 7–13, 2021 (online meeting)”

[14] Ree, Rimhak Lie elements and an algebra associated with shuffles, Ann. of Math. (2), Volume 68 (1958), pp. 210-220 | MR | Zbl

[15] Reutenauer, Christophe Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press, Oxford University Press, New York, 1993, xviii+269 pages (Oxford Science Publications)

[16] Schützenberger, Marcel-Paul Sur une propriété combinatoire des demi-groupes libres, C. R. Acad. Sci. Paris, Volume 245 (1957), pp. 16-18 | Zbl

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