nLab
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.

There is a classical survey of combinatorial Hopf algebras

• 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 the chapter 5 of S. Majid’s Foundation of quantum group theory.

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