Advertisement

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

Original Paper
• 12 Downloads

## 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

17B05 17D99

## References

1. 1.
Alexeevsky, D., Guha, P.: On decomposability of Nambu–Poisson tensor. Acta. Math. Univ. Comenian 65, 1–9 (1996)
2. 2.
Bai, C., Guo, L., Ni, X.: Generalizations of the classical Yang–Baxter equation and O-operators. J. Math. Phys. 52, 063515 (2011)
3. 3.
Bagger, J., Lambert, N.: Gauge symmetry and supersymmetry of multiple M2-branes. Phys. Rev. D. 77, 065008 (2008)
4. 4.
Bagger, J., Lambert, N.: Gauge symmetry and supersymmetry of multiple M2-branes. Phys. Rev. D 77, 065008 (2008)
5. 5.
Bai, C., Bellier, O., Guo, L., Ni, X.: Spliting of operations, Manin products and Rota-Baxter operators. IMRN.
6. 6.
Bai, R., Guo, L., Li, J.: Rota–Baxter 3-Lie algebras. J. Math. Phys. 54(6), 063504 (2013)
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)
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)
10. 10.
Filippov, V.T.: $$n$$-Lie algebras. Sib. Mat. Zh. 26, 126–140 (1985)
11. 11.
Guo, L., Zhang, B.: Renormalization of multiple zeta values. J. Algebra 319, 3770–3809 (2008)
12. 12.
Ho, P., Hou, R., Matsuo, Y.: Lie 3-algebra and multipleM2-branes. JHEP 0806, 020 (2008)
13. 13.
Manchon, D., Paycha, S.: Nested sums of symbols and renormalised multiple zeta values. Int. Math. Res. Pap. 24, 4628–4697 (2010)
14. 14.
Rota, G.C.: Baxter algebras and combinatorial identities I, II. Bull. Am. Math. Soc. 75(325–329), 330–334 (1969)
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)

## 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