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Deformations and Their Controlling Cohomologies of \({\mathcal{O}}\)-Operators

  • Rong Tang
  • Chengming Bai
  • Li GuoEmail author
  • Yunhe Sheng
Article
  • 4 Downloads

Abstract

\({\mathcal{O}}\)-operators are important in broad areas of mathematics and physics, such as integrable systems, the classical Yang–Baxter equation, pre-Lie algebras and splitting of operads. In this paper, a deformation theory of \({\mathcal{O}}\)-operators is established in consistence with the general principles of deformation theories. On the one hand, \({\mathcal{O}}\)-operators are shown to be characterized as the Maurer–Cartan elements in a suitable graded Lie algebra. A given \({\mathcal{O}}\)-operator gives rise to a differential graded Lie algebra whose Maurer–Cartan elements characterize deformations of the given \({\mathcal{O}}\)-operator. On the other hand, a Lie algebra with a representation is identified from an \({\mathcal{O}}\)-operator T such that the corresponding Chevalley–Eilenberg cohomology controls deformations of T, thus can be regarded as an analogue of the André–Quillen cohomology for the \({\mathcal{O}}\)-operator. Thereafter, linear and formal deformations of \({\mathcal{O}}\)-operators are studied. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order n deformations of an \({\mathcal{O}}\)-operator are also characterized in terms of the new cohomology theory. Applications are given to deformations of Rota–Baxter operators of weight 0 and skew-symmetric r-matrices for the classical Yang–Baxter equation. For skew-symmetric r-matrices, there is an independent Maurer–Cartan characterization of the deformations as well as an analogue of the André–Quillen cohomology, which turn out to have an explicit relationship with the ones obtained as \({\mathcal{O}}\)-operators associated to the coadjoint representations. Finally, linear deformations of skew-symmetric r-matrices and their corresponding triangular Lie bialgebras are studied.

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Acknowledgements

This research is supported by NSFC (11471139, 11425104, 11771190) and NSF of Jilin Province (20170101050JC). C. Bai is also supported by the Fundamental Research Funds for the Central Universities and Nankai ZhiDe Foundation.

References

  1. 1.
    Bai C.: Bijective 1-cocycles and classification of 3-dimensional left-symmetric algebras. Commun. Algebra 37, 1016–1057 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bai C.: A unified algebraic approach to the classical Yang–Baxter equation. J. Phys. A: Math. Theor. 40, 11073–11082 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bai, C., Bellier, O., Guo, L., Ni, X.: Spliting of operations, Manin products and Rota–Baxter operators. Int. Math. Res. Not., 485–524 (2013)Google Scholar
  4. 4.
    Balavoine, D.: Deformations of algebras over a quadratic operad. Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math. Amer. Math. Soc., Providence, vol. 202, 207–234 (1997)Google Scholar
  5. 5.
    Baxter G.: An analytic problem whose solution follows from a simple algebraic identity. Pac. J. Math. 10, 731–742 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baxter R.J.: One-dimensional anisotropic Heisenberg chain. Ann. Phys. 70, 323–337 (1972)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Baues O.: Left-symmetric algebras for \({\mathfrak{gl}({\rm n})}\). Trans. Am. Math. Soc. 351, 2979–2996 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bordemann M.: Generalized Lax pairs, the modified classical Yang–Baxter equation, and affine geometry of Lie groups. Commun. Math. Phys. 135, 201–216 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Burde D.: Simple left-symmetric algebras with solvable Lie algebra. Manuscr. Math. 95, 397–411 (1998)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Burde D.: Left-symmetric algebras, or pre-Lie algebras in geometry and physics. Cent. Eur. J. Math. 4, 323–357 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carinena J., Grabowski J., Marmo G.: Quantum bi-Hamiltonian systems. Internat. J. Modern Phys. A. 15(30), 4797–4810 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chapoton F., Livernet M.: Pre-Lie algebras and the rooted trees operad. Int. Math. Res. Not. 8, 395–408 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chari V., Pressley A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  14. 14.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dorfman I.: Dirac Structures and Integrability of Nonlinear Evolution Equations. Wiley, Chichester (1993)Google Scholar
  16. 16.
    Dzhumadil’daev A.: Cohomologies and deformations of right-symmetric algebras. J. Math. Sci. 93, 836–876 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Etingof P., Kazhdan D.: Quantization of Lie bialgebras. I. Sel. Math. (N.S.) 2, 1–41 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fox T.F.: An introduction to algebraic deformation theory. J. Pure Appl. Algebra 84, 17–41 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gerstenhaber M.: The cohomology structure of an associative ring. Ann. Math. 78, 267–288 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gerstenhaber M.: On the deformation of rings and algebras. Ann. Math. (2) 79, 59–103 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gerstenhaber M.: On the deformation of rings and algebras. II. Ann. Math. 84, 1–19 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gerstenhaber M.: On the deformation of rings and algebras. III. Ann. Math. 88, 1–34 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gerstenhaber M.: On the deformation of rings and algebras. IV. Ann. Math. 99, 257–276 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Guo, L.: An introduction to Rota–Baxter algebra. Surveys of Modern Mathematics, 4. International Press, Somerville, MA; Higher Education Press, Beijing, xii+226 pp (2012)Google Scholar
  25. 25.
    Hartshore R.: Deformation Theory, Graduate Texts in Math. 257. Springer, Berlin (2010)Google Scholar
  26. 26.
    Kodaira K., Spencer D.: On deformations of complex analytic structures I and II. Ann. Math. 67, 328–466 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kontsevich M.: Operads and motives in deformation quantization. Lett. Math. Phys. 48, 35–72 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66, 157–216 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kontsevich, M., Soibelman, Y.: Deformation theory. I [Draft] (2010) http://www.math.ksu.edu/~soibel/Book-vol1.ps
  30. 30.
    Kosmann-Schwarzbach Y., Magri F.: Poisson-Lie groups and complete integrability, I: drinfeld bialgebras, dual extensions and their canonical representations. Ann. Inst. H. Poincaré Phys. Théor. 49, 433–460 (1988)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Kupershmidt B.A.: What a classical r-matrix really is. J. Nonlinear Math. Phys. 6, 448–488 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Loday, J.-L.: Scindement d’associativté et algébres de Hopf. in the proceedings of conference in honor of Jean Leray, Nantes (2002), Séminaire et Congrés (SMF), 9, pp. 155–172 (2004)Google Scholar
  33. 33.
    Loday J.-L., Vallette B.: Algebraic Operads. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  34. 34.
    Mazur B.: Perturbations, deformations, and variations (and “near-misses”) in geometry, physics, and number theory. Bull. AMS 41, 307–336 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Nijenhuis, A.: Sur une classe de proprits communes quelques types differents d’algebres. (French) Enseignement Math. (2). 14 1968 225–277 (1970)Google Scholar
  36. 36.
    Nijenhuis A., Richardson R.: Cohomology and deformations in graded Lie algebras. Bull. Am. Math. Soc. 72, 1–29 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Nijenhuis A., Richardson R.: Commutative algebra cohomology and deformations of Lie and associative algebras. J. Algebra 9, 42–105 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Pei J., Bai C., Guo L.: Splitting of operads and Rota–Baxter operators on operads. Appl. Categor. Struct. 25, 505–538 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Reshetikhin N.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys. 20, 331–335 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Rieffel M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys. 122, 531–562 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Schomerus V.: D-branes and deformation quantization. J. High Energy Phys. 1999, 030 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Semonov-Tian-Shansky M.A.: What is a classical R-matrix? Funct. Anal. Appl. 17, 259–272 (1983)Google Scholar
  43. 43.
    Uchino K.: Twisting on associative algebras and Rota–Baxter type operators. J. Noncommut. Geom. 4, 349–379 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Voronov Th: Higher derived brackets and homotopy algebras. J. Pure Appl. Algebra 202, 133–153 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Wang, Q., Sheng, Y., Bai, C., Liu, J.: Nijenhuis operators on pre-Lie algebras. Commun. Contemp. Math. (2018).  https://doi.org/10.1142/S0219199718500505
  46. 46.
    Yang C.N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett. 19, 1312–1315 (1967)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsJilin UniversityChangchunChina
  2. 2.Chern Institute of Mathematics and LPMCNankai UniversityTianjinChina
  3. 3.Department of MathematicsJiangxi Normal UniversityNanchangChina
  4. 4.Department of Mathematics and Computer ScienceRutgers UniversityNewarkUSA

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