The geometric constraints on Filippov algebroids

Yanhui Bi; Zhixiong Chen; Zhuo Chen; Maosong Xiang; Yanhui Bi; Zhixiong Chen; Zhuo Chen; Maosong Xiang

doi:10.3934/math.2024539

AIMS Mathematics

2024, Volume 9, Issue 5: 11007-11023. doi: 10.3934/math.2024539

Previous Article Next Article

Research article Special Issues

The geometric constraints on Filippov algebroids

1.
Center for Mathematical Sciences, Nanchang Hangkong University, Nanchang, China
2.
College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang, China
3.
Department of Mathematical Sciences, Tsinghua University, Beijing, China
4.
Center for Mathematical Sciences, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China

Received: 15 January 2024 Revised: 02 March 2024 Accepted: 11 March 2024 Published: 20 March 2024
MSC : 53C15, 57R22

Filippov $ n $-algebroids are introduced by Grabowski and Marmo as a natural generalization of Lie algebroids. On this note, we characterized Filippov $ n $-algebroid structures by considering certain multi-input connections, which we called Filippov connections, on the underlying vector bundle. Through this approach, we could express the $ n $-ary bracket of any Filippov $ n $-algebroid using a torsion-free type formula. Additionally, we transformed the generalized Jacobi identity of the Filippov $ n $-algebroid into the Bianchi-Filippov identity. Furthermore, in the case of rank $ n $ vector bundles, we provided a characterization of linear Nambu-Poisson structures using Filippov connections.
- Filippov algebroids,
- anchored bundles,
- Filippov connections,
- Bianchi-Filippov identity,
- Nambu-Poisson structures
Citation: Yanhui Bi, Zhixiong Chen, Zhuo Chen, Maosong Xiang. The geometric constraints on Filippov algebroids[J]. AIMS Mathematics, 2024, 9(5): 11007-11023. doi: 10.3934/math.2024539

Related Papers:

Abstract

Filippov $ n $-algebroids are introduced by Grabowski and Marmo as a natural generalization of Lie algebroids. On this note, we characterized Filippov $ n $-algebroid structures by considering certain multi-input connections, which we called Filippov connections, on the underlying vector bundle. Through this approach, we could express the $ n $-ary bracket of any Filippov $ n $-algebroid using a torsion-free type formula. Additionally, we transformed the generalized Jacobi identity of the Filippov $ n $-algebroid into the Bianchi-Filippov identity. Furthermore, in the case of rank $ n $ vector bundles, we provided a characterization of linear Nambu-Poisson structures using Filippov connections.

References

[1]	S. Basu, S. Basu, A. Das, G. Mukherjee, Nambu structures and associated bialgebroids, Proc. Math. Sci., 129 (2019), 12. http://dx.doi.org/10.1007/s12044-018-0455-7 doi: 10.1007/s12044-018-0455-7
[2]	Y. Bi, J. Li, Higher Dirac structures and Nambu-Poisson geometry, Adv. Math., (Chinese), 52 (2023), 867–882.
[3]	R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, Bianchi identities for non-geometric fluxes from quasi-Poisson structures to Courant algebroids, Fortschr. Phys., 60 (2012), 1217–1228. https://dx.doi.org/10.1002/prop.201200099 doi: 10.1002/prop.201200099
[4]	J. P. Dufour, N. T. Zung, Linearization of Nambu structures, Compos. Math., 117 (1999), 83–105. http://dx.doi.org/10.1023/A:1001014910775 doi: 10.1023/A:1001014910775
[5]	V. T. Filippov, $n$-Lie algebras, Sib. Math. J., 26 (1985), 879–891. https://doi.org/10.1007/BF00969110 doi: 10.1007/BF00969110
[6]	K. Grabowska, J. Grabowski, Z. Ravanpak, VB-structures and generalizations, Ann. Global Anal. Geom., 62 (2022), 235–284. http://dx.doi.org/10.1007/s10455-022-09847-z doi: 10.1007/s10455-022-09847-z
[7]	J. Grabowski, G. Marmo, On Filippov algebroids and multiplicative Nambu-Poisson structures, Differ. Geom. Appl., 12 (2000), 35–50. http://dx.doi.org/10.1016/S0926-2245(99)00042-X doi: 10.1016/S0926-2245(99)00042-X
[8]	Y. Kosmann-Schwarzbach, K. Mackenzie, Differential operators and actions of Lie algebroids, arXiv: math/0209337, 2002. https://doi.org/10.48550/arXiv.math/0209337
[9]	C. Laurent-Gengoux, M. Stiénon, P. Xu, Poincaré-Birkhoff-Witt isomorphisms and Kapranov dg-manifolds, Adv. Math., 387 (2021), 107792. http://dx.doi.org/10.1016/j.aim.2021.107792 doi: 10.1016/j.aim.2021.107792
[10]	K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, 2010. http://dx.doi.org/10.1017/CBO9780511661839
[11]	K. C. H. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97–147. http://dx.doi.org/10.1112/blms/27.2.97 doi: 10.1112/blms/27.2.97
[12]	G. Marmo, G. Vilasi, A. M. Vinogradov, The local structure of n-Poisson and n-Jacobi manifolds, J. Geom. Phys., 25 (1998), 141–182. http://dx.doi.org/10.1016/S0393-0440(97)00057-0 doi: 10.1016/S0393-0440(97)00057-0
[13]	S. K. Mishra, G. Mukherjee, A. Naolekar, Cohomology and deformations of Filippov algebroids, Proc. Math. Sci., 132 (2022), 2. http://dx.doi.org/10.1007/s12044-021-00645-4 doi: 10.1007/s12044-021-00645-4
[14]	N. Nakanishi, On Nambu-Poisson manifolds, Rev. Math. Phys., 10 (1998), 499–510. http://dx.doi.org/10.1142/S0129055X98000161 doi: 10.1142/S0129055X98000161
[15]	Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D, 7 (1973), 2405–2412. http://dx.doi.org/10.1103/PhysRevD.7.2405 doi: 10.1103/PhysRevD.7.2405
[16]	P. Popescu, M. Popescu, Anchored vector bundles and Lie algebroids, In: Lie algebroids and related topics in differential geometry, 2000. http://dx.doi.org/10.4064/bc54-0-5
[17]	L. Takhtajan, On foundation of the generalized Nambu mechanics, Commun. Math. Phys., 160 (1994), 295–315.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)