The properties on F-manifold color algebras and pre-F-manifold color algebras

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Abstract

The concept of F-manifold algebras is an algebraic expression of F-manifolds. In this paper, we provide the definition of an F-manifold color algebra, which can be viewed as a natural generalization of an F-manifold algebra. We develop the representation theory of an F-manifold color algebra and show that F-manifold color algebras admitting non-degenerate symmetric bilinear forms are coherence F-manifold color algebras. The concept of pre-F-manifold color algebras is also presented, and using this definition one can construct F-manifold color algebras. These results extend some properties of F-manifold algebras.

Keywords

Lie color algebra / F-manifold algebra / F-manifold color algebra / representation theory / pre-F-manifold color algebra

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References

[1] B. Dubrovin, Geometry of 2D topological field theories, in Lecture Notes in Mathematics , 1620 (1996), Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094793

[2] C. Hertling, Y. I. Manin, Weak Frobenius manifolds, Int. Math. Res. Not., 1999 (1999), 277-286. https://doi.org/10.1155/S1073792899000148

[3] V. Dotsenko, Algebraic structures of F-manifolds via pre-Lie algebras , Ann. Mat. Pura Appl., 198 (2019), 517-527. https://doi.org/10.1007/s10231-018-0787-z

[4] C. Bai, Left-symmetric algebras from linear functions, J. Algebra, 281 (2004), 651-665. https://doi.org/10.1016/j.jalgebra.2004.06.036

[5] D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math., 4 (2006), 323-357. https://doi.org/10.2478/s11533-006-0014-9

[6] F. Chapoton, M. Livernet, Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not., 2001 (2001), 395-408. https://doi.org/10.1155/S1073792801000198

[7] S. Majid, W. Tao, Noncommutative differentials on Poisson-Lie groups and pre-Lie algebras, Pac. J. Math., 284 (2016), 213-256. https://doi.org/10.2140/pjm.2016.284.213

[8] J. Liu, Y. Sheng, C. Bai, F-manifold algebras and deformation quantization via pre-Lie algebras , J. Algebra, 559 (2020), 467-495. https://doi.org/10.1016/J.JALGEBRA.2020.04.029

[9] J. A. Cruz Morales, J. A. Gutierrez, A. Torres-Gomez, F-algebra-Rinehart pairs and super F-algebroids, São Paulo J. Math. Sci., 16 (2022), 727-742. https://doi.org/10.1007/s40863-021-00277-6

[10] Z. Chen, J. Li, M. Ding, On the classification of 3-dimensional F-manifold algebras, Czech. Math. J., 72 (2022), 1191-1204. https://doi.org/10.21136/CMJ.2022.0017-22

[11] A. B. Hassine, T. Chtioui, M. A. Maalaoui, S. Mabrouk, On Hom- F-manifold algebras and quantization , Turk. J. Math., 46 (2022), 1153-1176. https://doi.org/10.55730/1300-0098.3149

[12] M. Ding, Z. Chen, J. Li, F-manifold color algebras , preprint, arXiv:2101.00959. https://doi.org/10.48550/arXiv.2101.00959

[13] V. Rittenberg, D. Wyler, Generalized superalgebras, Nuclear Phys. B, 139 (1978), 189-202. https://doi.org/10.1016/0550-3213(78)90186-4

[14] M. Scheunert, Generalized Lie algebras, J. Math. Phys., 20 (1979), 712-720. https://doi.org/10.1063/1.524113

[15] J. Bergen, D. S. Passman, Delta ideals of Lie color algebras, J. Algebra, 177 (1995), 740-754. https://doi.org/10.1006/jabr.1995.1327

[16] L. Chen, Y. Ma, L. Ni, Generalized derivations of Lie color algebras, Results Math., 63 (2013), 923-936. https://doi.org/10.1007/s00025-012-0241-2

[17] X. Chen, S. D. Silvestrov, F. V. Oystaeyen, Representations and cocycle twists of color Lie algebras, Algebras Representation Theory, 9 (2006), 633-650. https://doi.org/10.1007/s10468-006-9027-0

[18] J. Feldvoss, Representations of Lie colour algebras, Adv. Math., 157 (2001), 95-137. https://doi.org/10.1006/aima.2000.1942

[19] Y. Ma, L. Chen, J. Lin, T^*-extension of Lie color algebras , Chin. Ann. Math., 35 (2014), 623-638.

[20] S. Wang, L. Zhu, Non-degenerate invariant bilinear forms on Lie color algebras, Algebra Colloq., 17 (2010), 365-374. https://doi.org/10.1142/S1005386710000362

[21] B. Sun, L. Chen, Y. Liu, T^*-extensions and abelian extensions of hom-Lie color algebras , Rev. Un. Mat. Argentina, 59 (2018), 123-142. https://doi.org/10.33044/revuma.v59n1a06

[22] M. Scheunert, R. Zhang, Cohomology of Lie superalgebras and their generalizations, J. Math. Phys., 39 (1998), 5024-5061. https://doi.org/10.1063/1.532508

[23] X. Ning, X. Wang, Lie color algebras and color left-symmetric structures, J. Math. (PRC), 27 (2007), 359-362. https://doi.org/10.13548/j.sxzz.2007.03.024