The infinitesimal Hopf algebra and the poset of planar forests
L. Foissy
DOI: 10.1007/s10801-008-0163-y
Abstract
We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A non-degenerate pairing and a dual basis are defined, and a combinatorial interpretation of the pairing in terms of orders on the vertices of planar forests is given. Moreover, the coproduct and the pairing can also be described with the help of a partial order on the set of planar forests, making it isomorphic to the Tamari poset. As a corollary, the dual basis can be computed with a Möbius inversion.
Pages: 277–309
Keywords: keywords infinitesimal Hopf algebra; planar tree; tamari poset
Full Text: PDF
References
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74. Cambridge University Press, Cambridge-New York (1980)
2. Aguiar, M.: Infinitesimal Hopf algebras. Contemp. Math. 267, 1-29 (2000)
3. Aguiar, M.: Infinitesimal bialgebras, pre-Lie and dendriform algebras. Lecture Notes in Pure and Appl. Math., vol.
237. Dekker, New York (2004)
4. Aguiar, M., Sottile, F.: Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. Adv. Math. 191(2), 225-275 (2005).
5. Aguiar, M., Sottile, F.: Structure of the Loday-Ronco Hopf algebra of trees. J. Algebra 295(2), 473- 511 (2006).
6. Connes, A., Kreimer, D.: Hopf algebras, Renormalization and Noncommutative geometry. Comm. Math. Phys 199(1), 203-242 (1998).
7. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem I. The Hopf algebra of graphs and the main theorem. Comm. Math. Phys. 210(1), 249-273 (2000).
8. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group. Comm. Math. Phys. 216(1), 215- 241 (2001).
9. Doi, Y.: Homological coalgebra. J. Math. Soc. Japan 33(1), 31-50 (1981)
10. Foissy, L.: Finite-dimensional comodules over the Hopf algebra of rooted trees. J. Algebra 255(1), 85-120 (2002).
11. Foissy, L.: Les algèbres de Hopf des arbres enracinés, I. Bull. Sci. Math. 126, 193-239 (2002)
12. Foissy, L.: Quantifications des algèbres de Hopf d'arbres plans décorés et lien avec les groupes quantiques. Bull. Sci. Math. 127(6), 505-548 (2003)
13. Grossman, R., Larson, R.G.: Hopf-algebraic structure of families of trees. J. Algebra 126(1), 184-210 (1989)
14. Holtkamp, R.: Comparison of Hopf Algebras on Trees. Arch. Math. (Basel) 80(4), 368-383 (2003)
15. Kreimer, D.: On the Hopf algebra structure of pertubative quantum field theories. Adv. Theor. Math. Phys. 2(2), 303-334 (1998).
16. Kreimer, D.: On Overlapping Divergences. Comm. Math. Phys. 204(3), 669-689 (1999).
17. Kreimer, D.: Combinatorics of (pertubative) Quantum Field Theory. Phys. Rep. 4-6, 387-424 (2002).
18. Loday, J.-L., Ronco, M.O.: Hopf algebra of the planar binary trees. Adv. Math. 139(2), 293-309 (1998)
19. Loday, J.-L., Ronco, M.O.: On the structure of cofree hopf algebras. J. Reine Angew. Math. 592, 123-155 (2006)
20. Moerdijk, I.: On the Connes-Kreimer construction of Hopf algebras. Contemp. Math. 271, 311-321 (2001).