Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras. (English) Zbl 1351.81076
Summary: The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the “baby model” of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.{
©2016 American Institute of Physics}
©2016 American Institute of Physics}
MSC:
| 16T30 | Connections of Hopf algebras with combinatorics |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 05C05 | Trees |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
References:
| [1] | Aguiar, M., On the associative analog of Lie bialgebras, J. Algebra, 244, 492-532 (2001) · Zbl 0991.16033 · doi:10.1006/jabr.2001.8877 |
| [2] | Andrew, G. E.; Guo, L.; Keigher, W.; Ono, K., Baxter algebras and Hopf algebras, Trans. Am. Math. Soc., 355, 4639-4656 (2003) · Zbl 1056.16025 · doi:10.1090/S0002-9947-03-03326-9 |
| [3] | Atkinson, F. V., Some aspects of Baxter’s function equation, J. Math. Anal. Appl., 7, 1-30 (1963) · Zbl 0118.12903 · doi:10.1016/0022-247X(63)90075-1 |
| [4] | Bai, C., A unified algebraic approach to the classical Yang-Baxter equation, J. Phys. A: Math. Theor., 40, 11073-11082 (2007) · Zbl 1118.17008 · doi:10.1088/1751-8113/40/36/007 |
| [5] | Bai, C.; Bellier, O.; Guo, L.; Ni, X., Spliting of operations, Manin products and Rota-Baxter operators, Int. Math. Res. Not., 2013, 3, 485-524 (2013) · Zbl 1314.18010 · doi:10.1093/imrn/rnr266 |
| [6] | Baxter, G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10, 731-742 (1960) · Zbl 0095.12705 · doi:10.2140/pjm.1960.10.731 |
| [7] | Cartier, P., On the structure of free Baxter algebras, Adv. Math., 9, 253-265 (1972) · Zbl 0267.60052 · doi:10.1016/0001-8708(72)90018-7 |
| [8] | Connes, A.; Kreimer, D., Hopf algebras, renormalization and non-commutative geometry, Commun. Math. Phys, 199, 1, 203-242 (1998) · Zbl 0932.16038 · doi:10.1007/s002200050499 |
| [9] | Connes, A.; Kreimer, D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys., 210, 249-273 (2000) · Zbl 1032.81026 · doi:10.1007/s002200050779 |
| [10] | Ebrahimi-Fard, K.; Guo, L., Rota-Baxter algebras and dendriform algebras, J. Pure Appl. Algebra, 212, 320-339 (2008) · Zbl 1132.16032 · doi:10.1016/j.jpaa.2007.05.025 |
| [11] | Ebrahimi-Fard, K.; Guo, L., Quasi-shuffles, mixable shuffles and Hopf algebras, J. Algebraic Combinatorics, 24, 83-101 (2006) · Zbl 1103.16025 · doi:10.1007/s10801-006-9103-x |
| [12] | Ebrahimi-Fard, K.; Guo, L., Free Rota-Baxter algebras and rooted trees, J. Algebra Appl., 7, 167-194 (2008) · Zbl 1154.16026 · doi:10.1142/S0219498808002746 |
| [13] | Ebrahimi-Fard, K.; Guo, L.; Kreimer, D., Spitzer’s identity and the algebraic Birkhoff decomposition in pQFT, J. Phys. A: Math. Gen., 37, 11037-11052 (2004) · Zbl 1062.81113 · doi:10.1088/0305-4470/37/45/020 |
| [14] | Foissy, L., Les algèbres de Hopf des arbres enracinés décorés. I, Bull. Sci. Math., 126, 193-239 (2002) · Zbl 1013.16026 · doi:10.1016/S0007-4497(02)01108-9 |
| [15] | Foissy, L., “Introduction to Hopf algebra of rooted trees,” available at . · Zbl 1017.16031 |
| [16] | Gao, X., Guo, L., and Zhang, T., “Bialgebra and Hopf algebra structures on free Rota-Baxter algebras,” e-print . |
| [17] | Grossman, R.; Larson, R. G., Hopf-algebraic structure of families of trees, J. Algebra, 126, 1, 184-210 (1989) · Zbl 0717.16029 · doi:10.1016/0021-8693(89)90328-1 |
| [18] | Guo, L.; Keigher, W., Baxter algebras and shuffle products, Adv. Math., 150, 117-149 (2000) · Zbl 0947.16013 · doi:10.1006/aima.1999.1858 |
| [19] | Guo, L., An Introduction to Rota-Baxter Algebra (2012) · Zbl 1271.16001 |
| [20] | Guo, L., Operated semigroups, Motzkin paths and rooted trees, J. Algebraic Combinatorics, 29, 35-62 (2009) · Zbl 1227.05271 · doi:10.1007/s10801-007-0119-7 |
| [21] | Guo, L.; Paycha, S.; Zhang, B., Renormalization by Birkhoff-Hopf factorization and by generalized evaluators: A case study, Noncommutative Geometry, Arithmetic, and Related Topics, 183-211 (2011) · Zbl 1253.81118 |
| [22] | Guo, L.; Zhang, B., Renormalization of multiple zeta values, J. Algebra, 319, 3770-3809 (2008) · Zbl 1165.11071 · doi:10.1016/j.jalgebra.2008.02.003 |
| [23] | Hoffman, M. E., Combinatorics of rooted trees and Hopf algebras, Trans. Am. Math. Soc., 355, 3795-3811 (2003) · Zbl 1048.16023 · doi:10.1090/S0002-9947-03-03317-8 |
| [24] | Holtkamp, R., Comparison of Hopf algebras on trees, Arch. Math. (Basel), 80, 368-383 (2003) · Zbl 1056.16030 · doi:10.1007/s00013-003-0796-y |
| [25] | Joni, S.; Rota, G.-C., Coalgebras and bialgebras in combinatorics, Stud. Appl. Math., 61, 93-139 (1979) · Zbl 0471.05020 · doi:10.1002/sapm197961293 |
| [26] | Kassel, C., Quantum Groups, Graduate Texts in Mathematics, 155 (1994) |
| [27] | Kreimer, D., On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys., 2, 303-334 (1998) · Zbl 1041.81087 · doi:10.4310/ATMP.1998.v2.n2.a4 |
| [28] | Kreimer, D., On overlapping divergences, Commun. Math. Phys., 204, 669-689 (1999) · Zbl 0977.81091 · doi:10.1007/s002200050661 |
| [29] | Kreimer, D.; Panzer, E., Renormalization and Mellin transforms, Computer Algebra in Quantum Field Theory, XII, 195-223 (2013) · Zbl 1308.81137 |
| [30] | Kurosh, A. G., Free sums of multiple operator algebras, Sib. Math. J., 1, 62-70 (1960) · Zbl 0096.25304 |
| [31] | Loday, J.-L., Scindement d’associativté et algébres de Hopf, 9, 155-172 (2004) · Zbl 1073.16032 |
| [32] | Loday, J.-L.; Ronco, M. O., Hopf algebra of the planar binary trees, Adv. Math., 139, 2, 293-309 (1998) · Zbl 0926.16032 · doi:10.1006/aima.1998.1759 |
| [33] | Manchon, D., Hopf algebras in renormalisation, Handbook of Algebra, 5, 365-427 (2008) · Zbl 1215.81071 |
| [34] | Milnor, J. W.; Moore, J. C., On the structure of Hopf algebras, Ann. Math., 81, 1965, 211-264 · Zbl 0163.28202 · doi:10.2307/1970615 |
| [35] | Moerdijk, I., On the Connes-Kreimer construction of Hopf algebras, Contemp. Math., 271, 311-321 (2001) · Zbl 0987.16032 · doi:10.1090/conm/271/04360 |
| [36] | Rosenkranz, M., Gao, X., and Guo, L., “An algebraic study of multivariable integration and linear substitution,” e-print . · Zbl 07115743 |
| [37] | 37.G.-C.Rota, “Baxter algebras and combinatorial identities. I,” Bull. Am. Math. Soc.75, 325-329 (1969);G.-C.Rota, “Baxter algebras and combinatorial identities. II,” Bull. Am. Math. Soc.75, 330-334 (1969).10.1090/s0002-9904-1969-12158-0 · Zbl 0319.05008 |
| [38] | Rota, G.-C., Baxter operators, an introduction, Gian-Carlo Rota on Combinatorics (1995) · Zbl 0841.01031 |
| [39] | Sweedler, M., Hopf Algebras (1969) · Zbl 0194.32901 |
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