The theory of cohomologies on multiplicative Hom-$ \delta $-Jordan Lie color triple systems is given. As an application, deformations and extensions on the multiplicative Hom-$ \delta $-Jordan Lie color triple system are characterized in view of relevant cohomology.
Citation: Lili Ma, Qiang Li. Cohomology and its applications on multiplicative Hom-$ \delta $-Jordan Lie color triple systems[J]. AIMS Mathematics, 2024, 9(9): 25936-25955. doi: 10.3934/math.20241267
The theory of cohomologies on multiplicative Hom-$ \delta $-Jordan Lie color triple systems is given. As an application, deformations and extensions on the multiplicative Hom-$ \delta $-Jordan Lie color triple system are characterized in view of relevant cohomology.
| [1] | S. Helgason, Differential geometry, Lie groups, and symmetric spaces, New York: Academic Press, 1978. https://doi.org/10.1090/gsm/034 |
| [2] | S. Okubo, Introdution to Octonion and other non-associative algebras in Physics, Cambridge: Cambridge University Press, 1995. https://doi.org/10.1017/CBO9780511524479 |
| [3] | K. Yamaguti, On the cohomology space of Lie triple system, Kumamoto J. Sci. Ser. A, 5 (1960), 44–52. |
| [4] |
F. Kubo, Y. Taniguchi, A controlling cohomology of the deformation theory of Lie triple systems, J. Algebra, 278 (2004), 242–250. https://doi.org/10.1016/j.jalgebra.2004.01.005 doi: 10.1016/j.jalgebra.2004.01.005
|
| [5] | T. Zhang, Notes on cohomologies of Lie triple systems, J. Lie Theory, 24 (2014), 909–929. |
| [6] |
S. Okubo, Parastatistics as Lie supertriple systems, J. Math. Phys., 35 (1994), 2785–2803. https://doi.org/10.1063/1.530486 doi: 10.1063/1.530486
|
| [7] |
S. Okubo, N. Kamiya, Jordan-Lie superalgebra and Jordan-Lie triple system, J. Algebra, 198 (1997), 388–411. https://doi.org/10.1006/jabr.1997.7144 doi: 10.1006/jabr.1997.7144
|
| [8] |
S. Okubo, N. Kamiya, Quasi-classical Lie superalgebras and Lie supertriple systems, Commun. Algebra, 30 (2002), 3825–3850. https://doi.org/10.1081/AGB-120005822 doi: 10.1081/AGB-120005822
|
| [9] |
N. Kamiya, S. Okubo, A construction of simple Jordan superalgebra of $F$ type from a Jordan-Lie triple system, Ann. Mat. Pura. Appl. IV Ser., 181 (2002), 339–348. https://doi.org/10.1007/s102310100045 doi: 10.1007/s102310100045
|
| [10] |
L. Ma, L. Chen, On $\delta$-Jordan Lie triple systems, Linear Multilinear A., 65 (2017), 731–751. https://doi.org/10.1080/03081087.2016.1202184 doi: 10.1080/03081087.2016.1202184
|
| [11] |
S. Wang, X. Zhang, S. Guo, Derivations and deformations of $\delta$-Jordan Lie supertriple systems, Adv. Math. Phys., 2019 (2019), 3295462. https://doi.org/10.1155/2019/3295462 doi: 10.1155/2019/3295462
|
| [12] | F. Ammar, S. Mabrouk, A. Makhlouf, Cohomology and deformations of Hom-algebras, J. Lie Theory, 21 (2011), 813–836. |
| [13] |
L. Ma, L. Chen, J. Zhao, $\delta$-Hom-Jordan Lie superalgebras, Commun. Algebra, 46 (2018), 1668–1697. https://doi.org/10.1080/00927872.2017.1354008 doi: 10.1080/00927872.2017.1354008
|
| [14] |
D. Yau, On $n$-ary Hom-Nambu and Hom-Nambu-Lie algebras, J. Geom. Phys., 62 (2012), 506–522. https://doi.org/10.1016/j.geomphys.2011.11.006 doi: 10.1016/j.geomphys.2011.11.006
|
| [15] |
B. Sun, L. Chen, Y. Liu, $T^*$-extensions and abelian extensions of Hom-Lie color algebras, Rev. Union. Mat. Argent., 59 (2018), 123–142. https://doi.org/10.33044/revuma.v59n1a06 doi: 10.33044/revuma.v59n1a06
|
| [16] |
L. Chen, Y. Yi, M. Chen, Y. Tang, Cohomology and 1-parameter formal deformations of Hom-$\delta$-Lie triple systems, Adv. Appl. Clifford Algebras, 29 (2019), 63. https://doi.org/10.1007/s00006-019-0982-z doi: 10.1007/s00006-019-0982-z
|
| [17] |
Q. Li, L. Ma, Central extensions and Nijenhuis operators of Hom-$\delta$-Jordan Lie triple systems, Adv. Math. Phys., 2022 (2022), 2706774. https://doi.org/10.1155/2022/2706774 doi: 10.1155/2022/2706774
|
| [18] |
Y. Ma, L. Chen, J. Lin, Central extensions and deformations of Hom-Lie triple systems, Commun. Algebra, 46 (2018), 1212–1230. https://doi.org/10.1080/00927872.2017.1339063 doi: 10.1080/00927872.2017.1339063
|
| [19] |
Q. Li, L. Ma, Nijenhuis operators and abelian extensions of Hom-$\delta$-Jordan Lie supertriple systems, Mathematics, 11 (2023), 871. https://doi.org/10.3390/math11040871 doi: 10.3390/math11040871
|
| [20] |
D. Gaparayi, S. Attan, A. N. Issa, Hom-Lie-Yamaguti superalgebras, Korean J. Math., 27 (2019), 175–192. https://doi.org/10.11568/kjm.2019.27.1.175 doi: 10.11568/kjm.2019.27.1.175
|
| [21] |
J. Zhao, L. Chen, L. Ma, Representations and $T^*$-extensions of Hom-Jordan-Lie algebras, Commun. Algebra, 44 (2016), 2786–2812. https://doi.org/10.1080/00927872.2015.1065843 doi: 10.1080/00927872.2015.1065843
|
| [22] |
T. Zhang, J. Li, Representations and cohomologies of Hom-Lie-Yamaguti algebras with applications, Colloq. Math., 148 (2017), 131–155. https://doi.org/10.4064/cm6903-6-2016 doi: 10.4064/cm6903-6-2016
|
| [23] |
Q. Li, L. Ma, 1-parameter formal deformations and abelian extensions of Lie color triple systems, Electron. Res. Arch., 30 (2022), 2524–2539. https://doi.org/10.3934/era.2022129 doi: 10.3934/era.2022129
|
| [24] |
Q. Li, L. Ma, 1-parameter formal deformations, Nijenhuis operators and abelian extensions of $\delta$-Lie color triple systems, Topol. Appl., 328 (2023), 108458. https://doi.org/10.1016/j.topol.2023.108458 doi: 10.1016/j.topol.2023.108458
|
| [25] |
J. Feldvoss, Representations of Lie color algebras, Adv. Math., 157 (2001), 95–137. https://doi.org/10.1006/aima.2000.1942 doi: 10.1006/aima.2000.1942
|
Lili Ma, Qiang Li. Cohomology and its applications on multiplicative Hom-$ \delta $-Jordan Lie color triple systems[J]. AIMS Mathematics, 2024, 9(9): 25936-25955. doi: 10.3934/math.20241267
DownLoad: