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% leave as is
\issue{ACA conference 2025}
\articlehead{}
\titlehead{A generalization of Magnus duality}
\authorhead{Vu NGUYEN DINH}
\setcounter{page}{1}

\begin{document}

\title{A generalization of Magnus duality}

\author{Vu NGUYEN DINH \\
Department of General Education \\
University of Science and Technology of Hanoi \\
18 Hoang Quoc Viet, Cau Giay, Ha Noi, Viet Nam \\
\url{nguyen-dinh.vu@usth.edu.vn}}

\date{\today}

\maketitle

\begin{abstract}
Let $X$ be the finite graded set $X = B + Z$ (where $B = \{ b_1, \ldots, b_M \}$ and $Z = \{ z_1, \ldots, z_N \}$) and $\K$ being a fixed ring.  In this talk, we first review a Zinbiel bialgebra structure over the associative algebra $\ncp{\K}{X}$ and its graded dual studied in \cite{Sch1, Burgunder, Loday2}. We then use the concept of  the classical Lazard elimination to construct a $\K$-linear basis of  $\ncp{\K}{X}$ which is called Magnus basis (\cite{MKS}). As the main purpose, we will explain how to use these generalized bialgebra structures over $\ncp{\K}{X}$ to provide combinatorial tools in order to obtain the duality of Magnus basis.  We claim that the duality can be automatically approached to any graded set $X = B + Z$ where $B = \{b_{\gamma} \}_{\gamma \in \Gamma}$ and $Z = \{z_{\lambda} \}_{\lambda \in \Lambda}$ ($ \Gamma, \Lambda$ are nonempty index sets).  In case $X = B + Z$ where $B = \{x_0 \}$ and $Z = \{x _{\lambda} \}_{\lambda \in \Lambda}$ ($\Lambda$: a nonempty index set, for example $\N_+$), the Magnus duality was appeared in \cite{Naka}, Theorem 3.2 to derive a formula of Le-Murakami, Furusho type that expresses arbitrary coefficients of a group-like series $J  \in \ncs{\K}{x_0,x_1}$ in terms
of the “regular” coefficients of  $J$ (\cite{Naka}, Theorem 4.1).\\

This is based on join works with Prof. Gerard Duchamp and  Prof. Vincel Hoang Ngoc Minh (\cite{Vu}).

\end{abstract}

\begin{thebibliography}{99}
\bibitem{Burgunder}{E. Burgunder}, \textit{A symmetric version of Kontsevich graph complex and Leibniz homology},  J. Lie Theory, 20 (1) : 127-165, 2010.
		%
		\bibitem{Loday2} J. L. Loday, \textit{Generalized bialgebras and triples of operads}, {Asterisque} 320 (2008), x+116 pp.
		%
		\bibitem{MKS} W. Magnus, A. Karass, D. Solitar, \textit{Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations}, Dover Publications (1976). 
		%
		\bibitem{Naka} H. Nakamura, \textit{Demi-shuffle duals of Magnus polynomials in free associative algebra,} Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 929-939.
		%
		\bibitem{Sch1} M. P. Schüzenberger, \textit{Sur une propriété combinatoire des algèbres de Lie libres pouvant être utilisée dans un problème de mathéatiques appliqués}, Séminaire Dubreil–Jacotin Pisot (Algèbre et théorie des nombres) (1958/59).
		%
		\bibitem{Vu} Vu  NGUYEN DINH, \textit{Combinatorics of Lazard Elimination and Interactions},  These (2023), Université  Sorbonne Paris Nord.

\end{thebibliography}

\end{document}
