Advertisement

Structure on the Simple Canonical Nambu Rota–Baxter 3-Lie Algebra \(A_{\partial }\)

  • RuiPu BaiEmail author
  • Yue Ma
  • Chuangchuang Kang
Original Paper

Abstract

In this paper, we study the structure of simple canonical Nambu 3-Lie algebra \(A_{\partial }=\sum \nolimits _{m\in Z} F z\exp (mx) \oplus \sum \nolimits _{m\in Z}F y\exp (mx)\). We pay close attention to a special class of Rota–Baxter operators, which are k-order homogeneous Rota–Baxter operators R of weight 1 and weight 0 satisfying \(R(L_m)=f(m+k)L_{m+k}\), \(R(M_m)=g(m+k)M_{m+k}\) for all generators \(\{ L_m=z\exp (mx),\)\( M_m= y\exp (-mx)~~| ~~m\in Z\}\), where \(f, g : A_{\partial } \rightarrow F\) are functions and \(k\in Z\). We obtain that R is a k-order homogeneous Rota–Baxter operator on \(A_{\partial }\) of weight 1 with \(k\ne 0\) if and only if \(R=0\), and R is a 0-order homogeneous Rota–Baxter operator on \(A_{\partial }\) of weight 1 if and only if R is one of the ten possibilities described in Theorems 2.4 and 2.8; R is a k-order homogeneous Rota–Baxter operator on \(A_{\partial }\) of weight 0 with \(k\ne 0\) if and only if R satisfies Theorem 3.1; and R is a 0-order homogeneous Rota–Baxter operator on \(A_{\partial }\) of weight 0 if and only if R is one of the four possibilities described in Theorem 3.3

Keywords

3-Lie algebra Homogeneous Rota–Baxter operator Canonical Nambu 3-Lie algebra Rota–Baxter 3-algebra 

Mathematics Subject Classification

17B05 17D99 

Notes

Acknowledgements

The first author was supported in part by the Natural Science Foundation (11371245) and the Natural Science Foundation of Hebei Province (A2018201126).

References

  1. 1.
    Alexeevsky, D., Guha, P.: On decomposability of Nambu–Poisson tensor. Acta. Math. Univ. Comenian 65, 1–9 (1996)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bai, C., Guo, L., Ni, X.: Generalizations of the classical Yang–Baxter equation and O-operators. J. Math. Phys. 52, 063515 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bagger, J., Lambert, N.: Gauge symmetry and supersymmetry of multiple M2-branes. Phys. Rev. D. 77, 065008 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bagger, J., Lambert, N.: Gauge symmetry and supersymmetry of multiple M2-branes. Phys. Rev. D 77, 065008 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bai, C., Bellier, O., Guo, L., Ni, X.: Spliting of operations, Manin products and Rota-Baxter operators. IMRN.  https://doi.org/10.1093/imrn/rnr266
  6. 6.
    Bai, R., Guo, L., Li, J.: Rota–Baxter 3-Lie algebras. J. Math. Phys. 54(6), 063504 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bai, R., Li, Z., Wang, W.: Infnite-dimensional 3-Lie algebras and their connections to Harish–Chandra module. Front. Math. Chin. 12(3), 515–530 (2017)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bai, C., Guo, L., Sheng, Y.: Bialgebras, the classical Yang–Baxter equation and Manin triples for 3-Lie algebras. arXiv:1604.05996 (2016)
  9. 9.
    Cartier, P.: On the structure of free Baxter algebras. Adv. Math. 9, 253–265 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Filippov, V.T.: \(n\)-Lie algebras. Sib. Mat. Zh. 26, 126–140 (1985)zbMATHGoogle Scholar
  11. 11.
    Guo, L., Zhang, B.: Renormalization of multiple zeta values. J. Algebra 319, 3770–3809 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ho, P., Hou, R., Matsuo, Y.: Lie 3-algebra and multipleM2-branes. JHEP 0806, 020 (2008)CrossRefGoogle Scholar
  13. 13.
    Manchon, D., Paycha, S.: Nested sums of symbols and renormalised multiple zeta values. Int. Math. Res. Pap. 24, 4628–4697 (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Rota, G.C.: Baxter algebras and combinatorial identities I, II. Bull. Am. Math. Soc. 75(325–329), 330–334 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rota, G.C.: Baxter Operators, an Introduction. In: Kung, J.P.S. (ed.) Gian-Carlo Rota on Combinatorics, Introductory Papers and Commentaries. Birkhäuser, Boston (1995)Google Scholar
  16. 16.
    Takhtajan, L.: On foundation of the generalized Nambu mechanics. Commun. Math. Phys. 160, 295–315 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Key Laboratory of Machine Learning and Computational, Intelligence of Hebei Province, College of Mathematics and Information ScienceHebei UniversityBaodingPeople’s Republic of China
  2. 2.College of Mathematics and Information ScienceHebei UniversityBaodingPeople’s Republic of China

Personalised recommendations