Rota-Baxter Algebra

様々な分野に登場する Rota-Baxter algebra という代数的構造がある。 Run-Qiang Jian [Jia13; Jia14] によると, Rota-Baxter operator は, Baxter の確率論の研究 [Bax60] で様々な関係式を統一的に扱うものとして登場した。その後, そのような operator を持つ algebra が Rota [Rot69] により Baxter algebra と名付けられたため, 現在では Rota-Baxter algebra として呼ばれているようである。

Rota-Baxeter algebra については, Ebrahimi-Fard と Patras による survey [EP] がある。そこでは, Guo の本 [Guo12], Kung, Rota, Yan の [KRY09], そして Rota の [Rot95] が参照されている。

Run-Qiang Jian [Jia13; Jia14] によると, 以下のことに関係している。

• 組み合せ論 [Rot69; KGP07]
• Connes-Kreimer の perturbative quantum field theory における renormalization [EG07]
• pre-Poisson algebras [Agu00; LHB07]
• Loday algebras [Ebr02; EG08c]
• quasi-shuffle algebras [EG06]
• multiple zeta values [EG08b; GZ10]

Connes-Kreimer theory と関係あるということは, tree Hopf algebra と深い関係にあるということである。 Ebraihimi-Fard と Guo [EG08a; Guo09] や Aguiar と Moreira の[AM06] など。

一般化や変種も色々考えられている。

• Brzeziński [Brz16] の Rota-Baxeter systems
• Guo と Li [LG22] の braided Rota-Baxter algebra
• Semenov-Tian-Shansky [Sem83] の modified Rota-Baxter algebra
• Gao, Guo, Zhang [ZGG20] の matching Rota-Baxter algebra
• Guo [Guo12] の Rota-Baxter family algebra
• Lazarev, Sheng, Tang [LST23; LST21] の relative Rota-Baxter Lie algebra と homotopy relative Rota-Baxter algebra
• Bai, Guo, Ma [BGM] による Rota-Baxter antisymmetric infinitesimal bialgebra
• Rota-Baxter Lie algebra [Bor90; Kup99]
• Rota-Baxter Lie group [GLS21]
• Rota-Baxter Lie algebroid [GLS21]
• Rota-Baxter Lie groupoid [GLS21]
• Rota-Baxter pre-Lie algebra [Guo+]
• Rota-Baxter Lie \(2\)-algebra [ZL]

References

[Agu00]

Marcelo Aguiar. “Pre-Poisson algebras”. In: Lett. Math. Phys. 54.4 (2000), pp. 263–277. url: http://dx.doi.org/10.1023/A:1010818119040.

[AM06]

Marcelo Aguiar and Walter Moreira. “Combinatorics of the free Baxter algebra”. In: Electron. J. Combin. 13.1 (2006), Research Paper 17, 38 pp. (electronic). arXiv: math/0510169. url: http://www.combinatorics.org/Volume_13/Abstracts/v13i1r17.html.

[Bax60]

Glen Baxter. “An analytic problem whose solution follows from a simple algebraic identity”. In: Pacific J. Math. 10 (1960), pp. 731–742.

[BGM]

Chengming Bai, Li Guo, and Tianshui Ma. Bialgebras, Frobenius algebras and associative Yang-Baxter equations for Rota-Baxter algebras. arXiv: 2112.10928.

[Bor90]

Martin Bordemann. “Generalized Lax pairs, the modified classical Yang-Baxter equation, and affine geometry of Lie groups”. In: Comm. Math. Phys. 135.1 (1990), pp. 201–216. url: http://projecteuclid.org/euclid.cmp/1104201925.

[Brz16]

Tomasz Brzeziński. “Rota-Baxter systems, dendriform algebras and covariant bialgebras”. In: J. Algebra 460 (2016), pp. 1–25. arXiv: 1503.05073. url: https://doi.org/10.1016/j.jalgebra.2016.04.018.

[Ebr02]

K. Ebrahimi-Fard. “Loday-type algebras and the Rota-Baxter relation”. In: Lett. Math. Phys. 61.2 (2002), pp. 139–147. arXiv: math-ph/0207043. url: http://dx.doi.org/10.1023/A:1020712215075.

[EG06]

Kurusch Ebrahimi-Fard and Li Guo. “Mixable shuffles, quasi-shuffles and Hopf algebras”. In: J. Algebraic Combin. 24.1 (2006), pp. 83–101. arXiv: math/0506418. url: http://dx.doi.org/10.1007/s10801-006-9103-x.

[EG07]

Kurusch Ebrahimi-Fard and Li Guo. “Rota-Baxter algebras in renormalization of perturbative quantum field theory”. In: Universality and renormalization. Vol. 50. Fields Inst. Commun. Providence, RI: Amer. Math. Soc., 2007, pp. 47–105. arXiv: hep- th/0604116.

[EG08a]

Kurusch Ebrahimi-Fard and Li Guo. “Free Rota-Baxter algebras and rooted trees”. In: J. Algebra Appl. 7.2 (2008), pp. 167–194. arXiv: math/0510266. url: http://dx.doi.org/10.1142/S0219498808002746.

[EG08b]

Kurusch Ebrahimi-Fard and Li Guo. “Multiple zeta values and Rota-Baxter algebras”. In: Integers 8.2 (2008), A4, 18. arXiv: math/ 0601558.

[EG08c]

Kurusch Ebrahimi-Fard and Li Guo. “Rota-Baxter algebras and dendriform algebras”. In: J. Pure Appl. Algebra 212.2 (2008), pp. 320–339. arXiv: math/0503647. url: http://dx.doi.org/10.1016/j.jpaa.2007.05.025.

[EP]

Kurusch Ebrahimi-Fard and Frederic Patras. Rota-Baxter Algebra. The Combinatorial Structure of Integral Calculus. arXiv: 1304.1204.

[GLS21]

Li Guo, Honglei Lang, and Yunhe Sheng. “Integration and geometrization of Rota-Baxter Lie algebras”. In: Adv. Math. 387 (2021), Paper No. 107834, 34. arXiv: 2009 . 03492. url: https://doi.org/10.1016/j.aim.2021.107834.

[Guo+]

Shuangjian Guo, Yufei Qin, Kai Wang, and Guodong Zhou. Cohomology theory of Rota-Baxter pre-Lie algebras of arbitrary weights. arXiv: 2204.13518.

[Guo09]

Li Guo. “Operated semigroups, Motzkin paths and rooted trees”. In: J. Algebraic Combin. 29.1 (2009), pp. 35–62. arXiv: 0710.0429. url: https://doi.org/10.1007/s10801-007-0119-7.

[Guo12]

Li Guo. An introduction to Rota-Baxter algebra. Vol. 4. Surveys of Modern Mathematics. International Press, Somerville, MA; Higher Education Press, Beijing, 2012, pp. xii+226. isbn: 978-1-57146-253-4.

[GZ10]

Li Guo and Bin Zhang. “Polylogarithms and multiple zeta values from free Rota-Baxter algebras”. In: Sci. China Math. 53.9 (2010), pp. 2239–2258. arXiv: 1210.1818. url: http://dx.doi.org/10.1007/s11425-010-4044-1.

[Jia13]

Run-Qiang Jian. “From quantum quasi-shuffle algebras to braided Rota-Baxter algebras”. In: Lett. Math. Phys. 103.8 (2013), pp. 851–863. arXiv: 1302.4289. url: https://doi.org/10.1007/s11005-013-0619-4.

[Jia14]

Run-Qiang Jian. “Construction of Rota-Baxter algebras via Hopf module algebras”. In: Sci. China Math. 57.11 (2014), pp. 2321–2328. arXiv: 1307.6966. url: https://doi.org/10.1007/s11425-014-4845-8.

[KGP07]

Ebrahimi-Fard Kurusch, José M. Gracia-Bondía, and Frédéric Patras. “Rota-Baxter algebras and new combinatorial identities”. In: Lett. Math. Phys. 81.1 (2007), pp. 61–75. arXiv: math/0701031. url: http://dx.doi.org/10.1007/s11005-007-0168-9.

[KRY09]

Joseph P. S. Kung, Gian-Carlo Rota, and Catherine H. Yan. Combinatorics: the Rota way. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2009, pp. xii+396. isbn: 978-0-521-73794-4. url: https://doi.org/10.1017/CBO9780511803895.

[Kup99]

Boris A. Kupershmidt. “What a classical \(r\)-matrix really is”. In: J. Nonlinear Math. Phys. 6.4 (1999), pp. 448–488. arXiv: math/ 9910188. url: http://dx.doi.org/10.2991/jnmp.1999.6.4.5.

[LG22]

Yunnan Li and Li Guo. “Braided Rota-Baxter algebras, quantum quasi-shuffle algebras and braided dendriform algebras”. In: J. Algebra Appl. 21.7 (2022), Paper No. 2250134, 25. arXiv: 1901. 02843. url: https://doi.org/10.1142/S0219498822501341.

[LHB07]

Xiuxian Li, Dongping Hou, and Chengming Bai. “Rota-Baxter operators on pre-Lie algebras”. In: J. Nonlinear Math. Phys. 14.2 (2007), pp. 269–289. url: http://dx.doi.org/10.2991/jnmp.2007.14.2.9.

[LST21]

Andrey Lazarev, Yunhe Sheng, and Rong Tang. “Deformations and homotopy theory of relative Rota-Baxter Lie algebras”. In: Comm. Math. Phys. 383.1 (2021), pp. 595–631. arXiv: 2008.06714. url: https://doi.org/10.1007/s00220-020-03881-3.

[LST23]

Andrey Lazarev, Yunhe Sheng, and Rong Tang. “Homotopy relative Rota-Baxter lie algebras, triangular \(L_\infty \)-bialgebras and higher derived brackets”. In: Trans. Amer. Math. Soc. 376.4 (2023), pp. 2921–2945. arXiv: 2008.00059. url: https://doi.org/10.1090/tran/8844.

[Rot69]

Gian-Carlo Rota. “Baxter algebras and combinatorial identities. I, II”. In: Bull. Amer. Math. Soc. 75 (1969), 325–329; ibid. 75 (1969), pp. 330–334.

[Rot95]

Gian-Carlo Rota. “Baxter operators, an introduction”. In: Gian-Carlo Rota on combinatorics. Contemp. Mathematicians. Birkhäuser Boston, Boston, MA, 1995, pp. 504–512.

[Sem83]

M. A. Semenov-Tyan-Shanskiı̆. “What a classical \(r\)-matrix is”. In: Funktsional. Anal. i Prilozhen. 17.4 (1983), pp. 17–33.

[ZGG20]

Yi Zhang, Xing Gao, and Li Guo. “Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras”. In: J. Algebra 552 (2020), pp. 134–170. arXiv: 1909 . 10577. url: https://doi.org/10.1016/j.jalgebra.2020.02.011.

[ZL]

Shilong Zhang and Jiefeng Liu. Rota-Baxter Lie \(2\)-algebras. arXiv: 2203.03403.