Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RELATIVE ROTA-BAXTER SYSTEMS ON LEIBNIZ ALGEBRAS

  • Apurba Das (Department of Mathematics Indian Institute of Technology) ;
  • Shuangjian Guo (School of Mathematics and Statistics Guizhou University of Finance and Economics)
  • Received : 2021.11.07
  • Accepted : 2022.12.12
  • Published : 2023.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce relative Rota-Baxter systems on Leibniz algebras and give some characterizations and new constructions. Then we construct a graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter systems. This allows us to define a cohomology theory associated with a relative Rota-Baxter system. Finally, we study formal deformations and extendibility of finite order deformations of a relative Rota-Baxter system in terms of the cohomology theory.

Keywords

Acknowledgement

This work was financially supported by the NSF of China (No. 12161013) and the innovative exploration and new academic seedling project of Guizhou University of Finance and Economics (No. 2022XSXMA17).

References

  1. C. Bai, O. Bellier, L. Guo, and X. Ni, Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN 2013 (2013), no. 3, 485-524. https://doi.org/10.1093/imrn/rnr266
  2. D. Balavoine, Deformations of algebras over a quadratic operad, in Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), 207-234, Contemp. Math., 202, Amer. Math. Soc., Providence, RI, 1997. https://doi.org/10.1090/conm/202/02581
  3. G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960), 731-742. http://projecteuclid.org/euclid.pjm/1103038223 103038223
  4. A. Bloh, On a generalization of the concept of Lie algebra, Dokl. Akad. Nauk SSSR 165 (1965), 471-473.
  5. M. Bordemann and F. Wagemann, Global integration of Leibniz algebras, J. Lie Theory 27 (2017), no. 2, 555-567.
  6. T. Brzezinski, Rota-Baxter systems, dendriform algebras and covariant bialgebras, J. Algebra 460 (2016), 1-25. https://doi.org/10.1016/j.jalgebra.2016.04.018
  7. V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1995.
  8. 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), no. 1, 249-273. https://doi.org/10.1007/s002200050779
  9. S. Covez, The local integration of Leibniz algebras, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 1, 1-35. https://doi.org/10.5802/aif.2754
  10. A. Das, Deformations of associative Rota-Baxter operators, J. Algebra 560 (2020), 144-180. https://doi.org/10.1016/j.jalgebra.2020.05.016
  11. A. Das, Generalized Rota-Baxter systems, preprint (2020), arXiv: 2007.13652.
  12. B. Dherin and F. Wagemann, Deformation quantization of Leibniz algebras, Adv. Math. 270 (2015), 21-48. https://doi.org/10.1016/j.aim.2014.10.022
  13. Y. Fregier and M. Zambon, Simultaneous deformations of algebras and morphisms via derived brackets, J. Pure Appl. Algebra 219 (2015), no. 12, 5344-5362. https://doi.org/10.1016/j.jpaa.2015.05.018
  14. M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267-288. https://doi.org/10.2307/1970343
  15. M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59-103. https://doi.org/10.2307/1970484
  16. L. Guo, An introduction to Rota-Baxter algebra, Surveys of Modern Mathematics, 4, International Press, Somerville, MA, 2012.
  17. A. Kotov and T. Strobl, The embedding tensor, Leibniz-Loday algebras, and their higher gauge theories, Comm. Math. Phys. 376 (2020), no. 1, 235-258. https://doi.org/10.1007/s00220-019-03569-3
  18. J.-L. Loday, Une version non commutative des algebres de Lie: les algebres de Leibniz, Enseign. Math. (2) 39 (1993), no. 3-4, 269-293.
  19. J.-L. Loday, Dialgebras, in Dialgebras and related operads, 7-66, Lecture Notes in Math., 1763, Springer, Berlin, 2001. https://doi.org/10.1007/3-540-45328-8_2
  20. J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), no. 1, 139-158. https://doi.org/10.1007/BF01445099
  21. A. Nijenhuis and R. W. Richardson, Jr., Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc. 72 (1966), 1-29. https://doi.org/10.1090/S0002-9904-1966-11401-5
  22. A. Nijenhuis and R. W. Richardson, Jr., Commutative algebra cohomology and deformations of Lie and associative algebras, J. Algebra 9 (1968), 42-53. https://doi.org/10.1016/0021-8693(68)90004-5
  23. F. Panaite and F. Van Oystaeyen, Twisted algebras and Rota-Baxter type operators, J. Algebra Appl. 16 (2017), no. 4, 1750079, 18 pp. https://doi.org/10.1142/S0219498817500797
  24. J. Pei, C. Bai, and L. Guo, Splitting of operads and Rota-Baxter operators on operads, Appl. Categ. Structures 25 (2017), no. 4, 505-538. https://doi.org/10.1007/s10485-016-9431-5
  25. M. A. Semenov-Tyan-Shanskii, What is a classical r-matrix?, Funct. Anal. Appl. 17 (1983), 259-272. https://doi.org/10.1007/BF01076717
  26. T. Strobl and F. Wagemann, Enhanced Leibniz algebras: structure theorem and induced Lie 2-algebra, Comm. Math. Phys. 376 (2020), no. 1, 51-79. https://doi.org/10.1007/s00220-019-03522-4
  27. R. Tang, C. Bai, L. Guo, and Y. Sheng, Deformations and their controlling cohomologies of O-operators, Comm. Math. Phys. 368 (2019), no. 2, 665-700. https://doi.org/10.1007/s00220-019-03286-x
  28. R. Tang and Y. Sheng, Leibniz bialgebras, relative Rota-Baxter operators and the classical Leibniz Yang-Baxter equation, J. Noncommutative Geom. to appear.
  29. R. Tang, Y. Sheng, and Y. Zhou, Deformations of relative Rota-Baxter operators on Leibniz algebras, Int. J. Geom. Methods Mod. Phys. 17 (2020), no. 12, 2050174, 21 pp. https://doi.org/10.1142/S0219887820501741