On a conjecture on transposed Poisson $ n $-Lie algebras

Junyuan Huang; Xueqing Chen; Zhiqi Chen; Ming Ding; Junyuan Huang; Xueqing Chen; Zhiqi Chen; Ming Ding

doi:10.3934/math.2024327

AIMS Mathematics

2024, Volume 9, Issue 3: 6709-6733. doi: 10.3934/math.2024327

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Research article

On a conjecture on transposed Poisson $ n $-Lie algebras

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2.
Department of Mathematics, University of Wisconsin-Whitewater, Whitewater WI.53190, USA
3.
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China

Received: 21 December 2023 Revised: 23 January 2024 Accepted: 31 January 2024 Published: 19 February 2024
MSC : 17A30, 17B63

The notion of a transposed Poisson $ n $-Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson $ n $-Lie algebra with a derivation gives rise to a transposed Poisson $ (n+1) $-Lie algebra. In this paper, we focus on transposed Poisson $ n $-Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of $ (n+1) $-Lie algebras from transposed Poisson $ n $-Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases.
- Lie algebra,
- Poisson algebra,
- transposed Poisson algebra,
- $ n $-Lie algebra,
- transposed Poisson $ n $-Lie algebra
Citation: Junyuan Huang, Xueqing Chen, Zhiqi Chen, Ming Ding. On a conjecture on transposed Poisson $ n $-Lie algebras[J]. AIMS Mathematics, 2024, 9(3): 6709-6733. doi: 10.3934/math.2024327

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Abstract

The notion of a transposed Poisson $ n $-Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson $ n $-Lie algebra with a derivation gives rise to a transposed Poisson $ (n+1) $-Lie algebra. In this paper, we focus on transposed Poisson $ n $-Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of $ (n+1) $-Lie algebras from transposed Poisson $ n $-Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases.

References

[1]	A. Agore, G. Militaru, Jacobi and Poisson algebras, J. Noncommut. Geom., 9 (2015), 1295–1342. https://dx.doi.org/10.4171/JNCG/224 doi: 10.4171/JNCG/224
[2]	C. Bai, R. Bai, L. Guo, Y. Wu, Transposed Poisson algebras, Novikov-Poisson algebras and $3$-Lie algebras, J. Algebra, 632 (2023), 535–566. https://dx.doi.org/10.1016/j.jalgebra.2023.06.006 doi: 10.1016/j.jalgebra.2023.06.006
[3]	R. Bai, Y. Wu, J. Li, H. Zhou, Constructing $(n + 1)$-Lie algebras from $n$-Lie algebras, J. Phys. A, 45 (2012), 475206. https://dx.doi.org/10.1088/1751-8113/45/47/475206 doi: 10.1088/1751-8113/45/47/475206
[4]	J. Bagger, N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D, 77 (2008), 065008. https://dx.doi.org/10.1103/PhysRevD.77.065008 doi: 10.1103/PhysRevD.77.065008
[5]	J. Bagger, N. Lambert, Three-algebras and $\mathcal{N} = 6$ Chern-Simons gauge theories, Phys. Rev. D, 79 (2009), 025002. https://dx.doi.org/10.1103/physrevd.79.025002 doi: 10.1103/physrevd.79.025002
[6]	P. D. Beites, B. L. M. Ferreira, I. Kaygorodov, Transposed Poisson structures, 2022. https://dx.doi.org/10.48550/arXiv.2207.00281
[7]	K. Bhaskara, K. Viswanath, Poisson algebras and Poisson manifolds, Longman Scientific & Technical, 1988.
[8]	Y. Billig, Towards Kac-van de Leur conjecture: Locality of superconformal algebras, Adv. Math., 400 (2022), 108295. https://dx.doi.org/10.1016/j.aim.2022.108295 doi: 10.1016/j.aim.2022.108295
[9]	N. Cantarini, V. Kac, Classification of linearly compact simple Jordan and generalized Poisson superalgebras, J. Algebra, 313 (2007), 100–124. https://dx.doi.org/10.1016/j.jalgebra.2006.10.040 doi: 10.1016/j.jalgebra.2006.10.040
[10]	N. Cantarini, V. Kac, Classification of linearly compact simple Nambu-Poisson algebras, J. Math. Phys., 57 (2016), 051701. https://dx.doi.org/10.1063/1.4948409 doi: 10.1063/1.4948409
[11]	V. Chari, A. Pressley, A guide to quantum groups, Cambridge University Press, 1995.
[12]	V. Dotsenko, Algebraic structures of $F$-manifolds via pre-Lie algebras, Ann. Mat. Pur. Appl., 198 (2019), 517–527. https://dx.doi.org/10.1007/s10231-018-0787-z doi: 10.1007/s10231-018-0787-z
[13]	A. S. Dzhumadil′daev, Identities and derivations for Jacobian algebras, 2002. https://dx.doi.org/10.48550/arXiv.math/0202040
[14]	B. L. M. Ferreira, I. Kaygorodov, V. Lopatkin, $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras, RACSAM, 115 (2021), 142. https://dx.doi.org/10.1007/s13398-021-01088-2 doi: 10.1007/s13398-021-01088-2
[15]	V. T. Filippov, $n$-Lie algebras, Sib. Math. J., 26 (1985), 126–140. https://dx.doi.org/10.1007/BF00969110 doi: 10.1007/BF00969110
[16]	M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math., 78 (1963), 267–288. https://dx.doi.org/10.2307/1970343 doi: 10.2307/1970343
[17]	Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Ann. I. Fourier, 46 (1996), 1243–1274. https://dx.doi.org/10.5802/aif.1547 doi: 10.5802/aif.1547
[18]	I. Laraiedh, S. Silvestrov, Transposed Hom-Poisson and Hom-pre-Lie Poisson algebras and bialgebras, 2021. https://dx.doi.org/10.48550/arXiv.2106.03277
[19]	G. Liu, C. Bai, A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras, 2023. https://dx.doi.org/10.48550/arXiv.2309.16174
[20]	J. Liu, C. Bai, Y. Sheng, Noncommutative Poisson bialgebras, J. Algebra, 556 (2020), 35–66. https://dx.doi.org/10.1016/j.jalgebra.2020.03.009 doi: 10.1016/j.jalgebra.2020.03.009
[21]	T. Ma, B. Li, Transposed BiHom-Poisson algebras, Commun. Algebra, 51 (2023), 528–551. https://dx.doi.org/10.1080/00927872.2022.2105343 doi: 10.1080/00927872.2022.2105343
[22]	Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D, 7 (1973), 2405–2412. https://dx.doi.org/10.1103/PhysRevD.7.2405 doi: 10.1103/PhysRevD.7.2405
[23]	X. Ni, C. Bai, Poisson bialgebras, J. Math. Phys., 54 (2013), 023515. https://dx.doi.org/10.1063/1.4792668 doi: 10.1063/1.4792668
[24]	A. Odzijewicz, Hamiltonian and quantum mechanics, Geom. Topol. Monogr., 17 (2011), 385–472.
[25]	A. Polishchuk, Algebraic geometry of Poisson brackets, J. Math. Sci., 84 (1997), 1413–1444. https://dx.doi.org/10.1007/BF02399197 doi: 10.1007/BF02399197
[26]	G. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc., 108 (1963), 195–222. https://dx.doi.org/10.2307/1993603 doi: 10.2307/1993603
[27]	L. Takhtajan, On foundation of the generalized Nambu mechanics, Commun. Math. Phys., 160 (1994), 295–315. https://dx.doi.org/10.1007/BF02103278 doi: 10.1007/BF02103278
[28]	X. Xu, Novikov-Poisson algebras, J. Algebra, 190 (1997), 253–279. https://dx.doi.org/10.1006/jabr.1996.6911 doi: 10.1006/jabr.1996.6911
[29]	L. Yuan, Q. Hua, $\frac{1}{2}$-(bi)derivations and transposed Poisson algebra structureson Lie algebras, Linear Multilinear A., 70 (2022), 7672–7701. https://dx.doi.org/10.1080/03081087.2021.2003287 doi: 10.1080/03081087.2021.2003287

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