Bimodules and matched pairs of noncommutative BiHom-(pre)-Poisson algebras

Abstract

The purpose of this paper is to introduce the notion of noncommutative BiHom-pre-Poisson algebra. Also, we establish the bimodules and matched pairs of noncommutative BiHom-(pre)-Poisson algebras. Their related relevant properties are also given. Finally, we exploit the notion of $\mathcal{O}$-operator to illustrate the relations existing between noncommutative BiHom-Poisson and noncommutative BiHom pre-Poisson algebras.

Keywords

• noncommutative BiHom-(pre)-Poisson algebras
• bimodules
• matched pairs
• $\mathcal{O}$-operator

References

1. [1] K. Abdaoui, F. Ammar and A. Makhlouf, Constructions and cohomology of Hom-Lie color algebras, Comm. Algebra 43 (11), 4581-4612, 2015.
2. [2] H. Adimi, H. Amri, S. Mabrouk and A. Makhlouf, (Non-BiHom-Commutative) BiHom-Poisson algebras, arXiv:2008.04597 [math.CT]
3. [3] M. Aguiar, Pre-Poisson algebras, Lett. Math. Phys. 54, 263-277, 2000.
4. [4] N. Aizawa and H. Sato, q-deformation of the Virasoro algebra with central extension, Phys. Lett. B 256, 185-190, 1991.
5. [5] F. Ammar and A. Makhlouf, Hom-Lie and Hom-Lie admissible superalgebras, J. Algebra 324, 1513-1528, 2010.
6. [6] S. Attan, Some characterizations of color Hom-Poisson algebras, Hacet. J. Math. Stat. 47 (6), 1552-1563, 2018.
7. [7] S. Attan, Structures and bimodules of simple Hom-alternative algebras, arXiv:1908.08711 [math.CT].
8. [8] S. Attan, H. Hounnon and B. Kpamègan, Hom-Jordan and Hom-alternative bimodules, Extracta Math. 35, 69-97, 2020.

9. [9] S. Attan and I. Laraiedh, Structures of BiHom-Poisson algebras, arXiv:2008.04763 [math.CT].
10. [10] I. Bakayoko, Modules over color Hom-Poisson algebras, J. of Gen. Lie Theory Appl. 8 (1), 2014.
11. [11] Y. Bao and Y. Ye, Cohomology structure for a Poisson algebra, I, J. Algebra Appl. 15 (2), 1650034, 2016.
12. [12] Y. Bao and Y. Ye, Cohomology structure for a Poisson algebra: II, Sci. China Math. 64 (5), 903920, 2019.
13. [13] A. Ben Hassine, T. Chtioui, S. Mabrouk and O. Ncib, Cohomology and linear deformation of BiHom-left-symmetric algebras, arXiv:1907.06979 [math.CT].
14. [14] M. Chaichian, D. Ellinas and Z. Popowicz, Quantum conformal algebra with central extension, Phys. Lett. B 248, 95-99, 1990.
15. [15] M. Chaichian, A.P. Isaev, J. Lukierski, Z. Popowic and P. Prešnajder, q-deformations of Virasoro algebra and conformal dimensions, Phys. Lett. B 262 (1), 32-38, 1991.
16. [16] M. Chaichian, P. Kulish and J. Lukierski, q-deformed Jacobi identity, q-oscillators and q-deformed infinite-dimensional algebras, Phys. Lett. B 237, 401-406, 1990.
17. [17] M. Chaichian and Z. Popowicz and P. Prešnajder, q-Virasoro algebra and its relation to the q-deformed KdV system, Phys. Lett. B 249, 63-65, 1990.
18. [18] T.L. Curtright and C.K. Zachos, Deforming maps for quantum algebras, Phys. Lett. B 243, 237-244, 1990.
19. [19] E.V. Damaskinsky and P.P. Kulish, Deformed oscillators and their applications, Zap. Nauch. Semin. LOMI 189, 37-74, 1991. [Engl. transl. in J. Sov. Math. 62, 2963-2986, 1992].
20. [20] C. Daskaloyannis, Generalized deformed Virasoro algebras, Modern Phys. Lett. A 7 (9), 809-816, 1992.
21. [21] G. Graziani, A. Makhlouf, C. Menini and F. Panaite, BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras, SIGMA Symmetry Integrability Geom. Methods Appl. 11, 2015.
22. [22] J.T. Hartwig, D. Larsson and S.D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra 295, 314-361, 2006.
23. [23] M.N. Hounkonnou, G.D. Houndedji and S. Silvestrov, Double constructions of biHom- Frobenius algebras, arXiv:2008.06645 [math.CT].
24. [24] N. Hu, q-Witt algebras, q-Lie algebras, q-holomorph structure and representations, Algebra Colloq. 6 (1), 51-70, 1999.
25. [25] C. Kassel, Cyclic homology of differential operators, the virasoro algebra and a qanalogue, Comm. Math. Phys. 146 (2), 343-356, 1992.
26. [26] F. Kubo, Finite-dimensional non-commutative Poisson algebras, Pure Appl. Algebra. 113, 307-314, 1996.
27. [27] F. Kubo, Non-commutative Poisson algebra structures on affine Kac-Moody aglebras, J. Pure Appl. Algebra. 126, 267-286, 1998.
28. [28] K. Kubo, Finite-dimensional simple Leibniz pairs and simple Poisson modules, Lett. Math. Phys. 43, 21-29, 1998.
29. [29] D. Larsson and S.D. Silvestrov, Quasi-hom-Lie algebras, central extensions and 2- cocycle-like identities, J. Algebra. 288, 321-344, 2005.
30. [30] J. Liu, C. Bai and Y. Sheng, Noncommutative Poisson bialgebras, arXiv:2004.02560 [math.CT].
31. [31] L. Liu, A. Makhlouf, C. Menini and F. Panaite, BiHom-pre-Lie algebras, BiHom- Leibniz algebras and Rota-Baxter operators on BiHom-Lie algebras, arXiv:1706.00474 [math.CT].
32. [32] L. Liu, A. Makhlouf, C. Menini and F. Panaite, Rota-Baxter operators on BiHomassociative algebras and related structures, arXiv:1703.07275 [math.CT].
33. [33] K.Q. Liu, Quantum central extensions, C. R. Math. Rep. Acad. Sci. Canada. 13 (4), 135-140, 1991.
34. [34] K.Q. Liu, Characterizations of the Quantum Witt Algebra, Lett. Math. Phys. 24 (4), 257-265, 1992.
35. [35] K.Q. Liu, The quantum Witt algebra and quantization of some modules over Witt algebra, PhD Thesis, Department of Mathematics, University of Alberta, Edmonton, Canada, 1992.
36. [36] X. Li, BiHom-Poisson algebra and its Application, Int. J. Alg. 13, 73-81, 2019.
37. [37] A. Makhlouf, Hom-dendriform algebras and Rota-Baxter Hom-algebras, in proceedings of international conferences in Nankai series in pure. In: C. Bai, L. Guo, J.-L. Loday, (eds.) Applied Mathematics and Theoretical Physics, World Scientific, Singapore, 9, 147-171, 2012.
38. [38] A. Makhlouf and S.D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2), 51-64, 2008.
39. [39] A. Makhlouf and S. Silvestrov, Hom-Lie Admissible Hom-Coalgebras and Hom-Hopf Algebras, Chapter 17, in: S. Silvestrov, E. Paal, V. Abramov, A. Stolin (Eds.), Generalized Lie theory in Mathematics, Physics and Beyond, Springer-Verlag, Berlin, Heidelberg, 189-206, 2009.
40. [40] A. Makhlouf and D. Yau, Rota-Baxter Hom-Lie admissible algebras, Comm. Alg. 23 (3), 1231-1257, 2014.
41. [41] M. Van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc. 360, 5711-5769, 2008.
42. [42] P. Xu, Noncommutative Poisson algebras, Amer. J. Math. 116, 101-125, 1994.
43. [43] D. Yau, Hom-algebras and homology, J. Lie Theory. 19 (2), 409-421, 2009.
44. [44] D. Yau, Hom-bialgebras and comodule Hom-algebras, Int. E. J. Alg. 8, 45-64, 2010.
45. [45] D. Yau, Hom-Malcev, Hom-alternative and Hom-Jordan algebras, Int. Elect. Journ. of Alg. 11, 177-217, 2012.
46. [46] D. Yau, A Hom-associative analogue of Hom-Nambu algebras, arXiv:1005.2373 [math.CT].
47. [47] D. Yau, Non-commutative Hom-Poisson algebras, arXiv:1010.3408[math.CT].
48. [48] L. Yuan, Hom-Lie color algebras, Comm. Alg. 40 (2), 575-592, 2012.