Tensor products and perturbations of BiHom-Novikov-Poisson algebras. (English) Zbl 1472.17098
A Novikov-Poisson algebra is given an associative product \(\cdot\) and a pre-Lie product \(*\), with the complementary axioms
\[
x*(y*z)=(x*z)*y,
\]
\[
(x*y)\cdot z-x*(y\cdot z)=(y*x)\cdot z-y*(x\cdot z),
\]
\[
(x\cdot y)*z=(x*z)\cdot y.
\]
A biHom Novikov-Poisson algebra is a Novikov-Poisson algebra with two twocommuting algebra endomorphisms, satisfying complementary conditions of compatibilities with the products \(\cdot\) and \(*\).
In this paper, a structure of biHom Novikov-Poisson algebra on the tensor product of two biHom Novikov-Poisson algebras. Moreover, it is shown that these objects can de deformed according to certain of their elements, generalizing a result due to Xu in the classical case and to Yau in the Hom case. Finally, a classe of biHom Novikov-Poisson algebras is introduced, which gives rise to biHom Poisson algebras. It is shown that this class is preserved under the tensor product, the Yau twist and the deformations mentioned above.
In this paper, a structure of biHom Novikov-Poisson algebra on the tensor product of two biHom Novikov-Poisson algebras. Moreover, it is shown that these objects can de deformed according to certain of their elements, generalizing a result due to Xu in the classical case and to Yau in the Hom case. Finally, a classe of biHom Novikov-Poisson algebras is introduced, which gives rise to biHom Poisson algebras. It is shown that this class is preserved under the tensor product, the Yau twist and the deformations mentioned above.
Reviewer: Loïc Foissy (Calais)
MSC:
| 17D30 | (non-Lie) Hom algebras and topics |
Keywords:
BiHom-Novikov-Poisson algebra; BiHom-Novikov algebra; BiHom-Poisson algebra; BiHom-commutative algebraReferences:
| [1] | Abramov, V.; Liivapuu, O.; Makhlouf, A., \( ( q , \sigma , \tau )\)-Differential graded algebras, Universe, 4, 12, 138 (2018) |
| [2] | Abramov, V.; Silvestrov, S., 3-Hom-Lie algebras based on \(\sigma \)-derivation and involution, Adv. Appl. Clifford Algebr., 30, 13 (2020), paper no. 45 · Zbl 1468.17031 |
| [3] | Adimi, H.; Amri, H.; Mabrouk, S.; Makhlouf, A., (Non-BiHom-commutative) BiHom-Poisson algebras (2020), Preprint |
| [4] | Aizawa, N.; Sato, H., \(q\)-Deformation of the Virasoro algebra with central extension, Phys. Lett. B, 256, 185-190 (1991) · Zbl 1332.17011 |
| [5] | Balinskii, A. A.; Novikov, S. P., Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras, Sov. Math. Dokl., 32, 228-231 (1985) · Zbl 0606.58018 |
| [6] | Caenepeel, S.; Goyvaerts, I., Monoidal Hom-Hopf algebras, Comm. Algebra, 39, 2216-2240 (2011) · Zbl 1255.16032 |
| [7] | Dubrovin, B. A.; Novikov, S. P., Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method, Sov. Math. Dokl., 27, 665-669 (1983) · Zbl 0553.35011 |
| [8] | Dubrovin, B. A.; Novikov, S. P., On Poisson brackets of hydrodynamic type, Sov. Math. Dokl., 30, 651-654 (1984) · Zbl 0591.58012 |
| [9] | Gel’fand, I. M.; Dorfman, I. Ya., Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl., 13, 248-262 (1979) · Zbl 0437.58009 |
| [10] | Graziani, G.; Makhlouf, A.; Menini, C.; Panaite, F., BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras, Symmetry Integrability Geom. Methods Appl., 11, 34 (2015), 086 · Zbl 1358.17006 |
| [11] | Guo, S.; Zhang, X.; Wang, S., The construction and deformation of BiHom-Novikov algebras, J. Geom. Phys., 132, 460-472 (2018) · Zbl 1442.17023 |
| [12] | Hartwig, J. T.; Larsson, D.; Silvestrov, S. D., Deformations of Lie algebras using \(\sigma \)-derivations, J. Algebra, 295, 314-361 (2006) · Zbl 1138.17012 |
| [13] | Hu, N., \(q\)-WItt algebras, \(q\)-Lie algebras, \(q\)-holomorph structure and representations, Algebra Colloq., 6, 51-70 (1999) · Zbl 0943.17007 |
| [14] | Larsson, D.; Silvestrov, S. D., Quasi-hom-Lie algebras, central extensions and \(2\)-cocycle-like identities, J. Algebra, 288, 321-344 (2005) · Zbl 1099.17015 |
| [15] | L. Liu, A. Makhlouf, C. Menini, F. Panaite, BiHom-pre-Lie algebras, BiHom-Leibniz algebras and Rota-Baxter operators on BiHom-Lie algebras, arXiv:math.QA/1706.00474. · Zbl 1509.17017 |
| [16] | Liu, L.; Makhlouf, A.; Menini, C.; Panaite, F., \( \{ \sigma , \tau \}\)-Rota-Baxter operators, infinitesimal hom-bialgebras and the associative (Bi)Hom-Yang-Baxter equation, Canad. Math. Bull., 62, 355-372 (2019) · Zbl 1460.17027 |
| [17] | Liu, L.; Makhlouf, A.; Menini, C.; Panaite, F., BiHom-Novikov algebras and infinitesimal BiHom-bialgebras, J. Algebra, 560, 1146-1172 (2020) · Zbl 1509.17017 |
| [18] | Liu, L.; Makhlouf, A.; Menini, C.; Panaite, F., Rota-Baxter operators on BiHom-associative algebras and related structures, Colloq. Math., 161, 263-294 (2020) · Zbl 1472.17097 |
| [19] | Makhlouf, A.; Silvestrov, S. D., Hom-algebras structures, J. Gen. Lie Theory Appl., 2, 51-64 (2008) · Zbl 1184.17002 |
| [20] | Makhlouf, A.; Silvestrov, S. D., Notes on formal deformations of Hom-associative and Hom-Lie algebras, Forum Math., 22, 715-759 (2010) · Zbl 1201.17012 |
| [21] | Xu, X., On simple Novikov algebras and their irreducible modules, J. Algebra, 185, 905-934 (1996) · Zbl 0863.17003 |
| [22] | Xu, X., Novikov-Poisson algebras, J. Algebra, 190, 253-279 (1997) · Zbl 0872.17030 |
| [23] | D. Yau, A twisted generalization of Novikov-Poisson algebras, arXiv:math.RA/1010.3410. |
| [24] | Yau, D., Hom-Novikov algebras, J. Phys. A., 44, Article 085202 pp. (2011) · Zbl 1208.81110 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
