Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-05-02T07:31:36.374Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPLEX PRODUCT STRUCTURES ON HOM-LIE ALGEBRAS

Published online by Cambridge University Press:  12 March 2018

L. NOURMOHAMMADIFAR  and
E. PEYGHAN
L. NOURMOHAMMADIFAR
Affiliation:
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran e-mails: [email protected], [email protected]
E. PEYGHAN
Affiliation:
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Iran e-mails: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we introduce the notion of complex product structures on hom-Lie algebras and show that a hom-Lie algebra carrying a complex product structure is a double hom-Lie algebra and it is also endowed with a hom-left symmetric product. Moreover, we show that a complex product structure on a hom-Lie algebra determines uniquely a left symmetric product such that the complex and the product structures are invariant with respect to it. Finally, we introduce the notion of hyper-para-Kähler hom-Lie algebras and we present an example of hyper-para-Kähler hom-Lie algebras.

Keywords

53C1553C2553D05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 1 , January 2019 , pp. 69 - 84
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Ammar, F., Ejbehi, Z. and Makhlouf, A., Cohomology and deformations of hom-algebras, J. Lie Theory 21 (4) (2011), 813–836.Google Scholar
2. Andrada, A., Complex product structures on 6-dimensional nilpotent Lie algebras, Forum Math. 20 (2008), 285–315.Google Scholar
3. Andrada, A. and Salamon, S., Complex product structures on Lie algebras, Forum Math. 17 (2005), 261–295.Google Scholar
4. Benayadi, S. and Boucetta, M., On para-Kähler and hyper-para-Kähler Lie algebras, J. Algebra 436 (2015), 61–101.Google Scholar
5. Caldarella, A. V., On paraquaternionic submersions between paraquaternionic Kähler manifolds, Acta Appl. Math. 112 (2010), 1–14.Google Scholar
6. Campoamor-Stursberg, R., Cardoso, I. E. and Ovando, G. P., Extending invariant complex structures, Int. J. Math. 26 (2015), 1–25.Google Scholar
7. Davidov, J., Grantcharov, G., Mushkarov, O. and Yotov, M., Compact complex surfaces with geometric structures related to split quaternions, Nucl. Phys. 865 (2012), 330–352.Google Scholar
8. Hartwig, J., Larsson, D. and Silvestrov, S., Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2006), 314–361.Google Scholar
9. Laurent-Gengoux, C., Makhlouf, A. and Teles, J., Universal algebra of a hom-Lie algebra and group-like elements, J. Pure Appl. Algebra 222 (5) (2018), 1139–1163.Google Scholar
10. Larsson, D. and Silvestrov, S., Quasi-hom-Lie algebras, central extensions and 2-cocycle-like identities, J. Algebra 288 (2005), 321–344.Google Scholar
11. Li, X., Hou, D. and Bai, C., Rota-Baxter operators on pre-Lie algebras, J. Nonlinear Math. Phys. 14 (2007), 269–289.Google Scholar
12. Majid, S., Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations, Pacific J. Math. 141 (1990), 311–332.Google Scholar
13. Peyghan, E. and Nourmohammadifar, L., Para-Kähler hom-Lie algebras, J. Algebra Appl. To appear.Google Scholar
14. Peyghan, E. and Nourmohammadifar, L., Complex and Kähler structures on hom-Lie algebras, arXiv:1610.07775.Google Scholar
15. Sheng, Y., Representations of hom-Lie algebras, Algebras Represent. Theor. 15 (2012), 1081–1098.Google Scholar
16. Sheng, Y. and Bai, C., A new approach to hom-Lie bialgebras, J. Algebra 399 (2014), 232–250.Google Scholar
17. Sheng, Y. and Chen, D., hom-Lie 2-algebras, J. Algebra 376 (2013), 174–195.Google Scholar
18. Yau, D., hom-algebras and homology, J. Lie Theory 19 (2009), 409–421.Google Scholar
19. Zhang, Y., Bai, C. and Guo, L., The category and operad of matching dialgebras, J. Algebra 21 (2013), 851–865.Google Scholar