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

Ming Ding; Zhiqi Chen; Jifu Li; Ming Ding; Zhiqi Chen; Jifu Li

doi:10.3934/era.2025005

Electronic Research Archive

2025, Volume 33, Issue 1: 87-101. doi: 10.3934/era.2025005

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Research article

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

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2.
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520, China
3.
School of Science, Tianjin University of Technology, Tianjin 300384, China

Received: 01 November 2024 Revised: 02 January 2025 Accepted: 07 January 2025 Published: 14 January 2025

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.
- Lie color algebra,
- F-manifold algebra,
- F-manifold color algebra,
- representation theory,
- pre-F-manifold color algebra
Citation: Ming Ding, Zhiqi Chen, Jifu Li. The properties on F-manifold color algebras and pre-F-manifold color algebras[J]. Electronic Research Archive, 2025, 33(1): 87-101. doi: 10.3934/era.2025005

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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.

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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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, S$\tilde{a}$o 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 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 doi: 10.13548/j.sxzz.2007.03.024

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