On differential Rota-Baxter algebras. (English) Zbl 1185.16038
Summary: A Rota-Baxter operator of weight \(\lambda\) is an abstraction of both the integral operator (when \(\lambda=0\)) and the summation operator (when \(\lambda=1\)). We similarly define a differential operator of weight \(\lambda\) that includes both the differential operator (when \(\lambda=0\)) and the difference operator (when \(\lambda=1\)). We further consider an algebraic structure with both a differential operator of weight \(\lambda\) and a Rota-Baxter operator of weight \(\lambda\) that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objects in the corresponding categories. In the commutative case, the free objects are given in terms of generalized shuffles, called mixable shuffles. In the noncommutative case, the free objects are given in terms of angularly decorated rooted forests. As a byproduct, we obtain structures of a differential algebra on decorated and undecorated planar rooted forests.
MSC:
| 16T30 | Connections of Hopf algebras with combinatorics |
| 16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |
| 05C05 | Trees |
Keywords:
differential operators; Rota-Baxter operators; differential Rota-Baxter algebras; free objects; categories of differential algebras; mixable shuffles; decorated rooted forestsReferences:
| [1] | Aguiar, M., On the associative analog of Lie bialgebras, J. Algebra, 244, 492-532 (2001) · Zbl 0991.16033 |
| [2] | Aguiar, M.; Moreira, W., Combinatorics of the free Baxter algebra, Electron. J. Combin., 13, 1, R17 (2006) · Zbl 1085.05006 |
| [3] | Baxter, G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 10, 731-742 (1960) · Zbl 0095.12705 |
| [4] | (Cassidy, P.; Guo, L.; Keigher, W.; Sit, W., Differential Algebra and Related Topics (Proceedings for the International Workshop in Newark, NJ, 2000) (2002), World Sci. Publishing) |
| [5] | Cohn, R. M., Difference Algebra (1965), Interscience Publishers · Zbl 0127.26402 |
| [6] | 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, Comm. Math. Phys., 210, 1, 249-273 (2000) · Zbl 1032.81026 |
| [7] | Connes, A.; Marcolli, M., From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory, (Frontiers in Number Theory, Physics, and Geometry (2006), Springer-Verlag), 617-713 · Zbl 1200.81113 |
| [8] | Diestel, R., Graph Theory (2005), Springer-Verlag, Available on-line: · Zbl 1074.05001 |
| [9] | Ebrahimi-Fard, K.; Guo, L., Quasi-shuffles, mixable shuffles and Hopf algebras, J. Algebraic Combin., 24, 83-101 (2006) · Zbl 1103.16025 |
| [10] | K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras and dendriform algebras, J. Pure Appl. Algebra (in press). arXiv:math.RA/0503647; K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras and dendriform algebras, J. Pure Appl. Algebra (in press). arXiv:math.RA/0503647 · Zbl 1132.16032 |
| [11] | K. Ebrahimi-Fard, L. Guo, Free Rota-Baxter algebras and rooted trees. arXiv:math.RA/0510266; K. Ebrahimi-Fard, L. Guo, Free Rota-Baxter algebras and rooted trees. arXiv:math.RA/0510266 · Zbl 1154.16026 |
| [12] | K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras in renormalization of perturbative quantum field theory, in: Proceedings of the Workshop on Renormalization and Universality in Mathematical Physics, Fields Institute, October 18-22, 2005, pp. 47-105. arXiv:hep-th/0604116; K. Ebrahimi-Fard, L. Guo, Rota-Baxter algebras in renormalization of perturbative quantum field theory, in: Proceedings of the Workshop on Renormalization and Universality in Mathematical Physics, Fields Institute, October 18-22, 2005, pp. 47-105. arXiv:hep-th/0604116 · Zbl 1148.81018 |
| [13] | Ebrahimi-Fard, K.; Guo, L.; Kreimer, D., Spitzer’s Identity and the Algebraic Birkhoff Decomposition in pQFT, J. Phys. A, 37, 11037-11052 (2004) · Zbl 1062.81113 |
| [14] | Ebrahimi-Fard, K.; Guo, L.; Manchon, D., Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion, Comm. Math. Phys., 267, 821-845 (2006) · Zbl 1188.17020 |
| [15] | Grossman, R. L.; Larson, R. G., Differential algebra structures on families of trees, Adv. Appl. Math., 35, 97-119 (2005) · Zbl 1092.16023 |
| [16] | Guo, L.; Keigher, W., Baxter algebras and shuffle products, Adv. Math., 150, 117-149 (2000) · Zbl 0947.16013 |
| [17] | Guo, L.; Keigher, W., On free Baxter algebras: Completions and the internal construction, Adv. Math., 151, 101-127 (2000) · Zbl 0964.16027 |
| [18] | L. Guo, B. Zhang, Renormalization of multiple zeta values. arXiv:math.NT/0606076; L. Guo, B. Zhang, Renormalization of multiple zeta values. arXiv:math.NT/0606076 |
| [19] | Hoffman, M., Quasi-shuffle products, J. Algebraic Combin., 11, 49-68 (2000) · Zbl 0959.16021 |
| [20] | Keigher, W., On the ring of Hurwitz series, Comm. Algebra, 25, 1845-1859 (1997) · Zbl 0884.13013 |
| [21] | Kolchin, E., Differential Algebra and Algebraic Groups (1973), Academic Press: Academic Press New York · Zbl 0264.12102 |
| [22] | MacLane, S., Categories for the Working Mathematician (1971), Springer-Verlag: Springer-Verlag New York · Zbl 0232.18001 |
| [23] | M. Rosenkranz, G. Regensburger, Solving and factoring boundary problems for linear ordinary differential equations in differential algebra, Technical Report, SFB F1322, 2007; M. Rosenkranz, G. Regensburger, Solving and factoring boundary problems for linear ordinary differential equations in differential algebra, Technical Report, SFB F1322, 2007 · Zbl 1151.34008 |
| [24] | Rota, G., Baxter algebras and combinatorial identities I, Bull. Amer. Math. Soc., 5, 325-329 (1969) · Zbl 0192.33801 |
| [25] | Rota, G., Baxter operators, an introduction, (Kung, Joseph P. S., Gian-Carlo Rota on Combinatorics, Introductory Papers and Commentaries (1995), Birkhäuser: Birkhäuser Boston) · Zbl 0841.01031 |
| [26] | Singer, M.; van der Put, M., (Galois Theory of Difference Equations. Galois Theory of Difference Equations, Lecture Notes in Mathematics, vol. 1666 (1997), Springer) · Zbl 0930.12006 |
| [27] | Singer, M.; van der Put, M., Galois Theory of Linear Differential Equations (2003), Springer · Zbl 1036.12008 |
| [28] | E.W. Weisstein, “Tree”. From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/Tree.html; E.W. Weisstein, “Tree”. From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/Tree.html |
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