Representations and module-extensions of 3-hom-Lie algebras. (English) Zbl 1368.17003
Summary: In this paper, we study the representations and module-extensions of 3-hom-Lie algebras. We show that a linear map between 3-hom-Lie algebras is a morphism if and only if its graph is a hom subalgebra and show that the set of derivations of a 3-hom-Lie algebra is a Lie algebra. Moreover, we introduce the definition of \(T_\theta\)-extensions and \(T_\theta^\ast\)-extensions of 3-hom-Lie algebras in terms of modules, providing the necessary and sufficient conditions for a \(2k\)-dimensional metric 3-hom-Lie algebra to be isometric to a \(T_\theta^\ast\)-extension.
Keywords:
3-hom-Lie algebra; module-extension; \(T_\theta\)-extension; \(T_\theta^\ast\)-extension; representationReferences:
| [1] | Bagger, J.; Lambert, N., Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D, 77, 6 (2008), 065008, 6 pp |
| [2] | Bagger, J.; Lambert, N., Comments on multiple M2-branes, J. High Energy Phys., 2 (2008), 105, 15pp |
| [3] | Gustavsson, A., Algebraic structure on parallel M2-branes, Nuclear Phys. B, 811, 1-2, 66-76 (2009) · Zbl 1194.81205 |
| [4] | Hartwig, J.; Larsson, D.; Silvestrov, S., Deformations of Lie algebras using \(\sigma \)-derivations, J. Algebra, 295, 321-344 (2006) |
| [5] | Bai, R.; Wu, W.; Li, Y.; Li, Z., Module extensions of 3-Lie algebras, Linear Multilinear A, 60, 4, 433-447 (2012) · Zbl 1264.17002 |
| [6] | Hu, N., \(Q\)-Witt algebras, \(q\)-Lie algebras, \(q\)-holomorph structure and representations, Algebra Colloq., 6, 1, 51-70 (1999) · Zbl 0943.17007 |
| [7] | Sheng, Y., Representations of Hom-Lie algebras, Algebr. Represent. Theory, 15, 6, 1081-1098 (2012) · Zbl 1294.17001 |
| [8] | Yau, D., The Hom-Yang-Baxter equation and Hom-Lie algebras, J. Math. Phys., 52, 5 (2011), 053502, 19 pp · Zbl 1317.16032 |
| [9] | Yau, D., Hom-quantum groups: I. Quasi-triangular Hom-bialgebras, J. Phys. A, 45, 6 (2012), 065203, 23 pp · Zbl 1241.81110 |
| [10] | Ataguema, H.; Makhlouf, A.; Silvestrov, S., Generalization of n-ary Nambu algebras and beyond, J. Math. Phys., 50, 8 (2009), 083501, 15pp · Zbl 1328.17004 |
| [11] | Ammar, F.; Mabrouk, S.; Makhlouf, A., Representations and Cohomology of n-ary multiplicative Hom-Nambu-Lie algebras, J. Geom. Phys., 61, 10, 1898-1913 (2011) · Zbl 1258.17007 |
| [12] | Bordemann, M., Nondegenerate invariant bilinear forms on nonassociative algebras, Acta Math. Univ. Comenian. (N.S.), 66, 2, 151-201 (1997) · Zbl 1014.17003 |
| [13] | Ammar, F.; Mabrouk, S.; Makhlouf, A., Quadratic n-ary Hom-Nambu Algebras (2011) · Zbl 1258.17007 |
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.
