Ternary Nambu F -algebras: generalizing F -manifolds and Nambu-Poisson structures
DOI:
https://doi.org/10.17398/Keywords:
Ternary Nambu F -algebra, representation, relative Rota-Baxter operator, ternary pre-Nambu F -algebraAbstract
In this paper, we introduce the notion of ternary Nambu F -algebras, which extend both F -algebras (F -manifold algebras) and Nambu-Poisson algebras. Their representation theory is developed, with particular attention to the definition of dual representations, requiring additional conditions analogous to those in the binary setting. We also establish the concept of coherent ternary Nambu F -algebras and investigate their construction from underlying F -algebras. Moreover, we define and study relative Rota-Baxter operators associated with representations, and show how they naturally give rise to ternary pre-Nambu F -algebras.
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[1] E. Abdaoui, S. Mabrouk, A. Makhlouf, Rota-Baxter operators on pre-Lie superalgebras, Bull. Malays. Math. Sci. Soc. 42 (2019), 1567 – 1606.
[2] J. Arnlind, A. Makhlouf, S. Silvestrov, Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras, J. Math. Phys. 51 (2010), 043515, 11 pp.
[3] J.A. de Azcárraga, J.M. Izquierdo, Cohomology of Filippov algebras and an analogue of Whitehead’s lemma, J. Phys.: Conf. Ser. 175 (2009), 012001.
[4] R. Bai, L. Guo, J. Li, Y. Wu, Rota-Baxter 3-Lie algebras, J. Math. Phys. 54 (2013), 064504, 14 pp.
[5] C. Bai, L. Guo, Y. Sheng, Bialgebras, the classical Yang-Baxter equation and Manin triples for 3-Lie algebras, Adv. Theor. Math. Phys. 23 (2019), 27 – 74.
[6] R. Bai, C. Bai, J. Wang, Realizations of 3-Lie algebras, J. Math. Phys. 51 (2010), 063505, 12 pp.
[7] R. Bai, G. Song, Y. Zhang, On classification of n-Lie algebras, Front. Math. China 6 (2011), 581 – 606.
[8] A. Basu, J.A. Harvey, The M2-M5 brane system and a generalized Nahm’s equation, Nuclear Phys. B 713 (2005), 136 – 150.
[9] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960), 731 – 742.
[10] A. Ben Hassine, T. Chtioui, M.A. Maalaoui, S. Mabrouk, On Hom-F-manifold algebras and quantization, Turkish J. Math. 46 (2022), 1153 – 1176.
[11] V. Chari, A. Pressley, “A guide to quantum groups”, Cambridge University Press, Cambridge, 1994.
[12] A. Connes, 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), 249 – 273.
[13] L. David, I.A.B. Strachan, Compatible metrics on a manifold and nonlocal bi-Hamiltonian structures, Int. Math. Res. Not. 2004 (2004), 3533 – 3557.
[14] L. David, I.A.B. Strachan, Dubrovin’s duality for F -manifolds with eventual identities, Adv. Math. 226 (2011), 4031 – 4060.
[15] V. Dotsenko, Algebraic structures of F -manifolds via pre-Lie algebras, Ann. Mat. Pura Appl. (4) 198 (2019), 517 – 527.
[16] B. Dubrovin, On almost duality for Frobenius manifolds, in “Geometry, topology, and mathematical physics”, Amer. Math. Soc. Transl. Ser. 2, 212, American Mathematical Society, Providence, RI, 2004, 75 – 132.
[17] A.S. Dzhumadil’daev, Representations of vector product n-Lie algebras, Comm. Algebra 32 (2004), 3315 – 3326.
[18] V.T. Filippov, n-Lie algebras, Sibirsk. Mat. Zh. 26 (1985), 126 – 140, 191.
[19] P. Gautheron, Some remarks concerning Nambu mechanics, Lett. Math. Phys. 37 (1996), 103 – 116.
[20] F. Harrathi, S. Sendi, O-operators, r-matrices and splitting of operations for non-commutative ternary Nambu-Poisson algebras, J. Geom. Phys. 217 (2025), Paper No. 105608, 20 pp.
[21] C. Hertling, “Frobenius manifolds and moduli spaces for singularities”, Cambridge Tracts in Math., 151, Cambridge University Press, Cambridge, 2002.
[22] C. Hertling, Y. Manin, Weak Frobenius manifolds, Internat. Math. Res. Notices 1999 (1999), 277 – 286.
[23] B.A. Kupershmidt, What a classical r-matrix really is, J. Nonlinear Math. Phys. 6 (1999), 448 – 488.
[24] P. Lorenzoni, M. Pedroni, A. Raimondo, F -manifolds and integrable systems of hydrodynamic type, Arch. Math. (Brno) 47 (2011), 163 – 180.
[25] S.M. Kasymov, On a theory of n-Lie algebras, Algebra i Logika 26 (1987), 277–297; English translation: Algebra and Logic 26 (1987), 155 – 166.
[26] Y.P. Lee, Quantum K-theory. I. Foundations, Duke Math. J. 121 (2004), 389 – 424.
[27] A. Makhlouf, H. Amri, Non-commutative ternary Nambu-Poisson algebras and ternary Hom-Nambu-Poisson algebras, J. Gen. Lie Theory Appl. 9 (2015), Article ID 1000221, 8 pp.
[28] S.A. Merkulov, Operads, deformation theory and F -manifolds, in “Frobenius manifolds.”, Aspects Math., E36, Friedr. Vieweg & Sohn, Wiesbaden, 2004, 213 – 251.
[29] D. Ming, Z. Chen, J. Li, F -manifold color algebras. arXiv:2101.00959
[30] Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D (3) 7 (1973), 2405 – 2412.
[31] A.P. Pozhidaev, n-ary Mal’tsev algebras, Algebra Logic 40 (2001), 170 – 182.
[32] M.A. Semenov-Tian-Shansky, What is a classical r-matrix?, Funktsional. Anal. i Prilozhen. 17 (1983), 17 – 33.
[33] L. Takhtajan, On foundation of the generalized Nambu mechanics, Comm. Math. Phys. 160 (1994), 295 – 315.
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