combinatorial Hopf algebra

Some Hopf algebras 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 denote either informal notion of Hopf algebra coming from combinatorics (and, in particular, study of various rings of symmetric functions), but also, now standard notion as a pair of a Hopf algebra $H$ over $k$ equipped with a character $\zeta: H\to k$, the zeta function of a Hopf algebra.

- Federico Ardila,
*Hopf algebras and combinatorics*, 30 lectures, youtube list

There is a classical survey

- Gian-Carlo Rota,
*Hopf algebra methods in combinatorics*, Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), pp. 363–365, Colloq. Internat. CNRS**260**, CNRS, Paris 1978.

and for the basic intuition for Hopf algebras in ocmbinatorics see the chapter 5 of S. Majid’s *Foundation of quantum group theory*. Nowdays the major book on combinatorial Hopf algebras is

- Marcelo Aguiar, Swapneel Mahajan,
*Monoidal Functors, Species and Hopf Algebras*, With forewords by Kenneth Brown, Stephen Chase and André Joyal. CRM Monograph Series**29**Amer. Math. Soc. 2010. lii+784 pp. (pdf draft)

Other references include

- G-C. Rota, J. A. Stein,
*Plethystic Hopf algebras*, Proc. Nat. Acad. Sci. U.S.A.**91**(1994), no. 26, 13057–13061, MR96e:16054,*Plethystic algebras and vector symmetric functions*, Proc. Nat. Acad. Sci. U.S.A.**91**(1994), no. 26, 13062–13066, MR96e:16055 - Kurusch Ebrahimi-Fard,
*Combinatorial Hopf algebras+, lectures at CIMAT summer school, slides pdf* - F. Hivert, J.-C. Novelli, J.-Y. Thibon,
*Trees, functional equations and combinatorial Hopf algebras*, Europ. J. Comb.**29**(1) (2008), 1682–1695. - Mercedes H. Rosas, Gian-Carlo Rota, Joel Stein,
*A combinatorial overview of the Hopf algebra of MacMahon symmetric functions*, Ann. Comb.**6**(2002), no. 2, 195–207. - Bertfried Fauser,
*On the Hopf algebraic origin of Wick normal ordering*, J. Phys. A 34 (2001), no. 1, 105–115 - Damien Calaque, Kurusch Ebrahimi-Fard, Dominique Manchon,
*Two interacting Hopf algebras of trees: a Hopf-algebraic approach to composition and substitution of B-series*, Adv. in Appl. Math.**47**(2011), no. 2, 282–308 - K. Ebrahimi-Fard, Li Guo,
*Mixable shuffles, quasi-shuffles and Hopf algebras*, J. Algebraic Combin.**24**(2006), no. 1, 83–101, MR2007d:05152,doi - K. Ebrahimi-Fard, D. Kreimer,
*The Hopf algebra approach to Feynman diagram calculations*, J. Phys. A 38 (2005), no. 50, R385–R407, MR2006k:81266, doi - Alain Connes, Dirk Kreimer,
*Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem*, Comm. Math. Phys.**210**(2000), no. 1, 249–273, hep-th/9912092, MR2002f:81070, doi,*II. The $\beta$-function, diffeomorphisms and the renormalization group*, Comm. Math. Phys.**216**(2001), no. 1, 215–241; hep-th/0003188, MR2002f:81071, doi

Revised on October 17, 2013 05:42:44
by Zoran Škoda
(161.53.130.104)