A Page on Rota-Baxter Algebra


Summary: A Rota-Baxter algebra is an associative algebra with a linear operator that generalizes the algebra of continuous functions with the integral operator. More precisely, for a given commutative ring k and &lambda in k, a Rota-Baxter k-algebra (of weight &lambda) is a k-algebra R together with a k-linear operator P on R such that
P(x)P(y)=P(P(x)y) + P(xP(y)) + &lambdaP(xy)
(1)
for all x,y in R. Such a linear operator is called a Rota-Baxter operator (of weight &lambda). It is easy to see that the Rota-Baxter operator is an abstraction of the integration by parts formula. Note that the relation (1) still makes sense when the associative algebra R is replaced by a k-module with a bilinear binary operation, such as the Lie bracket. The study of Rota-Baxter algebra originated from the probability study of Glenn Baxter in 1960 and was developed further by Cartier and the school of Rota in the 1960s and 1970s. Independently, this structure appeared in the Lie algebra context as the operator form of the classical Yang-Baxter equation started in the 1980s. Since the late 1990s, Rota-Baxter algebra has experienced a quite remarkable renascence and found important theoretical developments and applications in mathematical physics, operads, number theory and combinatorics.
Some references on Rota-Baxter algebra and related topics (please e-mail Li Guo (liguo@rutgers.edu) to add references):
  • M. Aguiar, Prepoisson algebras, Letters Math. Phys. 54 (2000), 263-277.
  • M. Aguiar, Infinitesimal Hopf algebras, Contemporary Mathematics, 267 (2000), 1-29.
  • M. Aguiar, On the associative analog of Lie bialgebras, Journal of Algebra 244 (2001), 492-532.
  • M. Aguiar and J.-L. Loday, Quadri-algebras, J. Pure Applied Algebra 191 (2004), 205-221. ArXiv:math.QA/03090171.
  • G.E. Andrew, L. Guo, W. Keigher, K. Ono, Baxter algebras and Hopf algebras, Trans. Amer. Math. Soc. 355 (2003), 4639-4656.
  • F. V. Atkinson, Some aspects of Baxter's functional equation, J. Math. Anal. and Applications 7 (1963), 1--30.
  • C. Bai, L. Guo and X. Ni, Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras, to appear in Comm. Math. Phys. arXiv:0910.3262.
  • C. Bai, L. Guo and X. Ni, O-operators on associative algebras and associative Yang-Baxter equations, arXiv:0910.3261.
  • G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960), 731--742.
  • J. F. Carinena, K. Ebrahimi-Fard, H. Figueroa and J. M. Gracia-Bondia, Hopf algebras in dynamical systems theory, Int. J. Geom. Meth. Mod. Phys. 4 (2007) 577-646, http://xxx.lanl.gov/abs/math/0701010.
  • P. Cartier, On the structure of free Baxter algebras, Adv. Math. 9 (1972), 253-265.
  • P. Cassidy, L. Guo, W. Keigher, W. Sit, Differential Algebra and Related Topics, World Scientific Publishing Company, 2002.
  • E. Castillo and R. Diaz, Rota-Baxter Categories, Int. Electron. J. Algebra 5 (2009) 27-57, http://arxiv.org/abs/0706.1068.
  • A. Connes and D. 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), 249-273.
  • A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The $\beta$-function, diffeomorphisms and the renormalization group., Comm. Math. Phys. 216 , (2001), 215-241.
  • A. Connes and M. Marcolli, Renormalization and motivic Galois theory, arXiv:math.NT/0409306
  • A. Connes and M. Marcolli, From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory, arXiv:hep-th/0411114.
  • S. L. de Braganca, Finite dimensional Baxter algebras, Studies in Applied Math. LIV (1975), 75--89.
  • R. Diaz and M. Paez, An identity in Rota-Baxter algebras, Sem. Lothar. Combin. 57 (2007), Article B57b, 8pp, http://arxiv.org/abs/math/0612194.
  • A. Dimakis and F. Muller-Hoissen, An algebraic scheme associated with noncommutative KP hierarchy and some of its extensions, J.Phys. A 38 (2005) 5453-5506, http://arxiv.org/abs/nlin/0501003.
  • K. Ebrahimi-Fard, Loday-type algebras and the Rota--Baxter relation, Letters in Mathematical Physics 61 139, (2002).
  • K. Ebrahimi-Fard, On the associative Nijenhuis relation, Electrical J. Combin. 11 (1), (2004), R38. arXiv:math-ph/0302062.
  • K. Ebrahimi-Fard, J. M. Gracia-Bondia, L. Guo and J. C. Varilly, Combinatorics of renormalization as matrix calculus, Physics Letters B, 19 (2006), 552-558, arXiv:hep-th/0508154.
  • K. Ebrahimi-Fard, J. M. Gracia-Bondia and F. Patras, Lie theoretic approach to renormalization, Commun. Math. Phys. 276 (2007) 519-549, http://xxx.lanl.gov/abs/hep-th/0609035.
  • K. Ebrahimi-Fard, J. M. Gracia-Bondia and F. Patras, Rota-Baxter algebras and new combinatorial identities, Letters in Mathematical Physics 81 (2007) 61-75, http://xxx.lanl.gov/abs/math/0701031.
  • K. Ebrahimi-Fard and L. Guo, Quasi-shuffles, Mixable Shuffles and Hopf Algebras, J. Algebraic Combinatorics 24, (2006), 83-101, arXiv:math.RA/0506418.
  • K. Ebrahimi-Fard and L. Guo, On the products and dual of binary, quadratic, regular operads, J. Pure and Applied Algebra 200 (2005), 293-317, arXiv:math.RA/0407162.
  • K. Ebrahimi-Fard and L. Guo, Unit actions on operads and Hopf algebras, Theory and Applications of Categories 18 (2007), 348-371, arXiv:math.RA/0503342.
  • K. Ebrahimi-Fard and L. Guo, Rota--Baxter algebras and dendriform dialgebras, Jour. Pure Appl. Algebra 212 (2008), 320-339, arXiv:math/0503647.
  • K. Ebrahimi-Fard and L. Guo, On matrix representation of renormalization in prerturbative quantum field theory, arXiv:hep-th/0508155
  • K. Ebrahimi-Fard and L. Guo, Free Rota-Baxter algebras and rooted trees, (with K. Ebrahimi-Fard), J. Algebra and Its Applications, 7 (2008), 167-194.
  • K. Ebrahimi-Fard and L. Guo, Rota--Baxter algebras and multiple zeta values, Integers 8 (2008) A4.
  • K. Ebrahimi-Fard and L. Guo, Rota-Baxter algebras in renormalization of perturbative quantum field theory, In Universality and Renormalization, I. Binder and D. Kreimer, editors, Fields Institute Communicatins, v. 50, AMS 2007, 47-105, arXiv:hep-th/0604116.
  • K. Ebrahimi-Fard, L. Guo and D. Kreimer, Integrable Renormalization I: the Ladder Case, J. Math. Phys 45 (2004), 3758-3769. arXiv:hep-th/0402095.
  • K. Ebrahimi-Fard, L. Guo and D. Kreimer, Integrable Renormalization II: the General Case, Annales Henri Poincare 6 (2005) 369-395, arXiv:hep-th/0403118.
  • K. Ebrahimi-Fard, L. Guo and D. Kreimer, Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT, J. Phys. A: Math. Gen. 37 (2004) 11037-11052. arXiv:hep-th/0407082.
  • K. Ebrahimi-Fard and D. Kreimer, Hopf algebra approach to Feynman diagram calculations, J. Phys. A 38 (2005) R385-R406, http://xxx.lanl.gov/abs/hep-th/0510202.
  • K. Ebrahimi-Fard, L. Guo and D. Manchon Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion, Comm. Math. Physics 267 (2006) 821-845, arXiv:math-ph/0602004.
  • K. Ebrahimi-Fard and D. Manchon, Dendriform Equations, Jour. Algebra 322 (2009), 4053-4079, http://xxx.lanl.gov/abs/0805.0762.
  • K. Ebrahimi-Fard and D. Manchon, A Magnus- and Fer-type formula in dendriform algebras, Foundations of Computational Mathematics 9 (2009), 295, http://xxx.lanl.gov/abs/0707.0607.
  • K. Ebrahimi-Fard and D. Manchon, On matrix differential equations in the Hopf algebra of renormalization, Adv. Theor. Math. Phys. 10 (2006) 879-913, http://xxx.lanl.gov/abs/math-ph/0606039.
  • K. Ebrahimi-Fard and D. Manchon, The combinatorics of Bogoliubov's recursion in renormalization, CIRM 2006 workshop "Renormalization and Galois Theory", Org. F. Fauvet, J.-P. Ramis, http://xxx.lanl.gov/abs/0710.3675.
  • K. Ebrahimi-Fard, D. Manchon and F. Patras, New identities in dendriform algebras, Jour. Algebra 320, 708-727 (2008), http://xxx.lanl.gov/abs/0705.2636.
  • B. Fauser and P. Jarvis, The Dirichlet Hopf algebra of arithmetics, J. of Knot Theor. and its Ramifications 16 (2007) 379-438, http://arxiv.org/abs/math-ph/0511079.
  • J. M. Freeman, On the classification of operator identities, Studies in Appl. Math. 51 (1972), 73--84.
  • J. M. Freeman, Spitzer's formula for non-commutative algebras}, Proceedings of the 3rd Southeastern Conf. on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1972), 205--211, (1972).
  • L. Guo, Properties of free Baxter algebras,} Adv. Math. 151 (2000), 346-- 374.
  • L. Guo, Baxter algebras and the umbral calculus, Adv. in Appl. Math., 27 (2001), 405-426.
  • L. Guo, Ascending chain conditions in free Baxter algebras,} Internat. J. Algebra Comput., 12 (2002), 601-622.
  • L. Guo, Baxter algebra and differential algebra}, in Differential Algebra and Related Topics, World Scientific Publishing Company, 2002, 281-305.
  • L. Guo, Baxter algebras, Stirling numbers and partitions}, J. Alg. Appl. 4 (2005), 153-164, arXiv:math.AC/0402348.
  • L. Guo, Algebraic Birkhoff decomposition and its applications, Automorphic Forms and the Langlands Program, International Press, 2008, 283-323.
  • L. Guo, Operated semigroups, Motzkin paths and rooted trees, J. Algebraic Combinatorics 29 (2009), 35-62.
  • L. Guo, WHAT IS a Rota-Baxter algebra, Notice of Amer. Math. Soc. 56, (2009) 1436-1437.
  • L. Guo and W. Keigher, On integration algebras, unpublished manuscript, 1998.
  • L. Guo, W. Keigher, Baxter algebras and shuffle products, Adv. Math. 150 (2000), 117--149.
  • L. Guo, W. Keigher, On free Baxter algebras: completions and the internal construction, Adv. Math. 151 (2000), 101-127.
  • L. Guo and W. Keigher, On differential Rota-Baxter algebras, J. Pure Appl. Algebra 212 (2008) 522-540, arXiv:math.RA/0703780.
  • L. Guo and Z. Liu, Rota-Baxter operators on generalized power series rings, to appear in J. Algebra Its Applications arXiv:0710.0433v1 [math.RA].
  • L. Guo, S. Paycha, B. Xie and B. Zhang, Double shuffle relations and renormalization of multiple zeta values, to appear in Proceedings of the Conference on Algebraic Cycles, 2008.
  • L. Guo, S. Paycha and B. Zhang, A comparison study of renormalization, preprint.
  • L. Guo and W. Sit, Generating functions and enumeration of Rota-Baxter words, (with William Y. Sit), in Proceedings ISSAC 2006, Genoa, Italy, ACM Press, 124-131, 2006, arXiv: math.RA/0602449.
  • L. Guo and W. Sit, Enumeration of differential Rota-Baxter words (with W. Sit), preprint.
  • L. Guo and B. Xie, Weighted sum formula for multiple zeta values, to appear in J. Number Theory, arXiv:0809.5110v1 [math.NT].
  • L. Guo and B. Xie, Explicit double shuffle relations and a generalization of Euler's decomposition formula, arXiv:0808.2618v1 [math.NT].
  • L. Guo and B. Xie, Structure theorems of mixable shuffle algebras and free commutative Rota-Baxter algebras, arXiv:0807.2267v2 [math.RA].
  • L. Guo and B. Zhang, Renormalization of multiple zeta values, J. Algebra 319 (2008), 3770-3809.
  • L. Guo and B. Zhang, Differential Birkhoff decomposition and renormalization of multiple zeta values, J. Number Theory 128 (2008) 2318-2339.
  • J. F. C. Kingman, Spitzer's identity and its use in probability thoery, J. London Math. Soc. 37 (1962), 309--316.
  • J. P. S. Kung, Baxter algebras, entry in Springer Online Encyclopaedia of Mathematics, M. Hazewinkel, ed. http://reference.springerlink.com/
  • P. Leroux, Ennea-algebras, J. Algebra 281 (2004), 287-302, ArXiv:math.QA/0309213.
  • P. Leroux, Construction of Nijenhuis operators and dendriform trialgebras, preprint, Nov. 2003, ArXiv:math.QA/0311132.
  • P. Leroux, On some remarkable operads constructed from Baxter operators, preprint, Nov. 2003, ArXiv:math.QA/0311214.
  • J.-L. Loday, La renaissance des operades, Seminaire Bourbaki, Vol. 1994/95. Asterisque No. 237 (1996), Exp. No. 792, 3, 47-74.
  • J.-L. Loday, Dialgebras, in Dialgebras and related operads, Lecture Notes in Math., 1763 (2002), 7-66. ArXiv:math.QA/0102053.
  • J.-L. Loday, Scindement d'associativite et algebres de Hopf. to appear in the Proceedings of the Conference in honor of Jean Leray, Nantes (2002), Seminaire et Congres (SMF) 9 (2004), 155-172.
  • L.-L. Loday, On the algebra of quasi-shuffles, preprint, arXiv:math.QA/0506498.
  • J.-L. Loday and M. O. Ronco, Hopf algebra of the planar binary trees, Adv. Math., 139, (1998), 293-309.
  • J.-L. Loday and M. Ronco, Trialgebras and families of polytopes,} in ``Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-theory" Contemporary Mathematics 346 (2004). ArXiv:math.AT/0205043.
  • D.~Manchon, Hopf algebras, from basics to applications to renormalization}, Comptes-rendus des Rencontres mathematiques de Glanon 2001. {arXiv:math.QA/0408405}
  • F. Menous, Formulas for Birkhoff-(Rota-Baxter) decompositions related to connected bialgebra, http://arxiv.org/abs/0710.0848.
  • J. B. Miller, Some properties of Baxter operators,} Acta Math. Acad. Sci. Hungar. 17 (1966), 387--400.
  • S. Moch, P. Uwer, S. Weinzierl, Nested sums, expansion of transcendental functions and multiple-loop integrals, J. Math. Phys. 43 (2002) no. 6, 3363-3386.
  • Nguyen-Huu-Bong, A formula for the resolvent of a summation operator, Nanta Math. 9 (1976), 106--108.
  • Nguyen-Huu-Bong, Some combinatorial properties of summation operators. J. Combinatorial Theory Ser. A 14 (1973), 253--255.
  • Nguyen-Huu-Bong, Some combinatorial properties of summation operators. J. Combinatorial Theory Ser. A 11 (1971), 213--221.
  • A. G. Reyman, M. A. Semenov-Tian-Shansky, Reduction of Hamitonian systems, affine Lie algebras and Lax equations. I, Invent. Math. 54 (1979) 81-100.
  • A. G. Reyman, M. A. Semenov-Tian-Shansky, Group theoretical methods in the theory of finite dimensional integrable systems}, in: Encyclopedia of mathematical science v.16: Dynamical Systems VII}, Springer (1994), 116-220.
  • S. Roman, G.-C. Rota, The umbral calculus, Adv. Math. 27 (1978), 95--188.
  • G.-C. Rota, Baxter algebras and combinatorial identities I, II,} Bull. Amer. Math. Soc. 75 (1969), 325--329, 330--334.
  • G.-C. Rota, Baxter operators, an introduction, In: ``Gian-Carlo Rota on Combinatorics, Introductory papers and commentaries", Joseph P.S. Kung, Editor, Birkhauser, Boston, 1995.
  • G.-C. Rota, Ten mathematics problems I will never solve, Invited address at the joint meeting of the American Mathematical Society and the Mexican Mathematical Society, Oaxaca, Mexico, December 6, 1997. DMV Mittellungen Heft 2, 1998, 45--52.
  • G.-C. Rota, D. A. Smith, Fluctuation theory and Baxter algebras, Istituto Nazionale di Alta Matematica, IX (1972), 179--201.
  • M. A. Semenov-Tian-Shansky, What is a classical $r$-matrix?, Funct. Ana. Appl., 17, no.4., (1983) 259-272.
  • M. A. Semenov-Tian-Shansky, Integrable Systems and Factorization Problems, Lectures given at the Faro International Summer School on Factorization and Integrable Systems (Sept. 2000), Sept. 2002, preprint: arXiv: nlin.SI/0209057.
  • F. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323--339.
  • G. P. Thomas, Frames, Young tableaux and Baxter sequences, Adv. Math. 26 (1977), 275--289.
  • K. Uchino, Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators, Lett. Math. Phys. 85 (2008) 91-109, http://arxiv.org/abs/math/0701320.
  • W.~Vogel, Die kombinatorische Losung einer Operator-Gleichung, Z. f. Wahrscheinlichkeitstheorie und verw. Gebiete, 2 , 122--134, (1963).
  • J. G. Wendel, A brief proof of a theorem of Baxter, Math. Scand. 11 (1962), 107--108.
  • R. Winkel, Sequences of symmetric polynomials and combatorial properties of tableaux, Adv. Math., 134 (1998), 46--89.