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\section*{combinatorial Hopf algebra}

Some [[Hopf algebra]]s encode relevant combinatorial information and are quite common and useful in [[algebraic combinatorics]] in general. More recently Hopf algebras appeared also in the study of [[renormalization]] of QFT, controlling its combinatorics. Combinatorial Hopf algebra may refer either to a Hopf algebra coming from combinatorics (and, in particular, to the study of various rings of symmetric functions), but also to the now standard notion of a pair, $(H,\zeta)$, consisting of a Hopf algebra $H$ over $k$ together with a character $\zeta: H\to k$, the zeta function of a Hopf algebra.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

One of the original sources is:

\begin{itemize}%
\item S. A. Joni and [[Gian-Carlo Rota]], \emph{Coalgebras and bialgebras in combinatorics}, Umbral Calculus and Hopf Algebras (Norman, OK, 1978), Contemp. Math., vol. 6, AMS, Providence, RI, 1982.

\end{itemize}
There is also another early article from Rota:

\begin{itemize}%
\item [[Gian-Carlo Rota]], \emph{Hopf algebra methods in combinatorics}, Probl\`e{}mes combinatoires et th\'e{}orie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), pp. 363--365, Colloq. Internat. CNRS \textbf{260}, CNRS, Paris 1978.

\end{itemize}
(Despite the title, though, this is just a short note that mainly discusses applications of Rota's theory to the so-called [[umbral calculus]], without making the Hopf algebra structure explicit.)

For the basic intuition for Hopf algebras in combinatorics see the chapter 5 of [[S. Majid]]`s \emph{Foundation of quantum group theory}. Nowadays the major book on combinatorial Hopf algebras is

\begin{itemize}%
\item [[Marcelo Aguiar]], Swapneel Mahajan, \emph{[[Monoidal Functors, Species and Hopf Algebras]]}, With forewords by [[Kenneth Brown]], Stephen Chase and [[André Joyal]]. CRM Monograph Series \textbf{29} Amer. Math. Soc. 2010. lii+784 pp. (\href{http://www.math.cornell.edu/~maguiar/a.pdf}{pdf draft})

\end{itemize}
Other references include:

\begin{itemize}%
\item G-C. Rota, J. A. Stein, \emph{Plethystic Hopf algebras}, Proc. Nat. Acad. Sci. U.S.A. \textbf{91} (1994), no. 26, 13057--13061, \href{http://www.ams.org/mathscinet-getitem?mr=1306900}{MR96e:16054}, \emph{Plethystic algebras and vector symmetric functions}, Proc. Nat. Acad. Sci. U.S.A. \textbf{91} (1994), no. 26, 13062--13066, \href{http://www.ams.org/mathscinet-getitem?mr=1306901}{MR96e:16055}
\item Kurusch Ebrahimi-Fard, \emph{Combinatorial Hopf algebras+, lectures at CIMAT summer school, slides \href{http://www-fourier.ujf-grenoble.fr/~peters/CIMAT/Abstracts/kurusch.pdf}{pdf}}
\item F. Hivert, J.-C. Novelli, J.-Y. Thibon, \emph{Trees, functional equations and combinatorial Hopf algebras}, Europ. J. Comb. \textbf{29} (1) (2008), 1682--1695.
\item Mercedes H. Rosas, Gian-Carlo Rota, Joel Stein, \emph{A combinatorial overview of the Hopf algebra of MacMahon symmetric functions}, Ann. Comb. \textbf{6} (2002), no. 2, 195--207.
\item Bertfried Fauser, \emph{On the Hopf algebraic origin of Wick normal ordering}, J. Phys. A 34 (2001), no. 1, 105--115
\item [[Damien Calaque]], Kurusch Ebrahimi-Fard, Dominique Manchon, \emph{Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series}, Adv. in Appl. Math. \textbf{47} (2011), no. 2, 282--308
\item K. Ebrahimi-Fard, Li Guo, \emph{Mixable shuffles, quasi-shuffles and Hopf algebras}, J. Algebraic Combin. \textbf{24} (2006), no. 1, 83--101, \href{http://www.ams.org/mathscinet-getitem?mr=2245782}{MR2007d:05152},\href{http://dx.doi.org/10.1007/s10801-006-9103-x}{doi}
\item K. Ebrahimi-Fard, D. Kreimer, \emph{The Hopf algebra approach to Feynman diagram calculations}, J. Phys. A 38 (2005), no. 50, R385--R407, \href{http://www.ams.org/mathscinet-getitem?mr=2199729}{MR2006k:81266}, \href{http://dx.doi.org/10.1088/0305-4470/38/50/R01}{doi}
\item [[Alain Connes]], [[Dirk Kreimer]], \emph{Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem}, Comm. Math. Phys. \textbf{210} (2000), no. 1, 249--273, \href{http://arxiv.org/abs/hep-th/9912092}{hep-th/9912092}, \href{http://www.ams.org/mathscinet-getitem?mr=1748177}{MR2002f:81070}, \href{http://dx.doi.org/10.1007/s002200050779}{doi}, \emph{II. The $\beta$-function, diffeomorphisms and the renormalization group}, Comm. Math. Phys. \textbf{216} (2001), no. 1, 215--241; \href{http://arxiv.org/abs/hep-th/0003188}{hep-th/0003188}, \href{http://www.ams.org/mathscinet-getitem?mr=1748177}{MR2002f:81071}, \href{http://dx.doi.org/10.1007/PL00005547}{doi}
\item [[Jean-Louis Loday]] and María O. Ronco. Combinatorial Hopf algebras. \emph{Quanta of maths}, Clay Math. Proc. 11, 347–383, Amer. Math. Soc., Providence, RI, 2010. \href{http://www-irma.u-strasbg.fr/~loday/PAPERS/2011LodayRonco%28CHA%29.pdf}{pdf}
\item Bill Schmitt, Hopf algebras of combinatorial structures, Canadian Journal of Mathematics 45 (1993), 412-428. \href{http://home.gwu.edu/~wschmitt/papers/hacs.pdf}{pdf}
\item Bill Schmitt, Hopf algebra methods in graph theory, Journal of Pure and Applied Algebra 101 (1995), 77-90. \href{http://home.gwu.edu/~wschmitt/papers/hamgt.pdf}{pdf}
\item Bill Schmitt, Incidence Hopf algebras, Journal of Pure and Applied Algebra 96 (1994), 299-330. \href{http://home.gwu.edu/~wschmitt/papers/iha.pdf}{pdf}

\end{itemize}
See also this nice lecture series on YouTube:

\begin{itemize}%
\item Federico Ardila, \emph{Hopf algebras and combinatorics}, 30 lectures, \href{http://www.youtube.com/playlist?list=PL-XzhVrXIVeRLeezwY9h4M68k6yB3yOo-}{youtube list}, with \href{http://math.sfsu.edu/federico/Clase/hopfindex.pdf}{index and overview} by Sara Billey.

\end{itemize}
[[!redirects combinatorial Hopf algebras]]



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