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. 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.

Literature

One of the original sources is:

• S. A. Joni and Gian-Carlo Rota, Coalgebras and bialgebras in combinatorics, Umbral Calculus and Hopf Algebras (Norman, OK, 1978), Contemp. Math., vol. 6, AMS, Providence, RI, 1982.

There is also another early article from Rota:

• 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.

(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 Foundation of quantum group theory. Nowadays the major book on combinatorial Hopf algebras is

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