Complex and Kähler structures on hom-Lie algebras

Abstract

(Almost) Complex and Hermitian structures on hom-Lie algebras are introduced and some examples of these structures are presented. We study the complexification of hom-Lie algebras. Also, the notion of Kähler hom-Lie algebras is introduced and then using a Kähler hom-Lie algebra, we present a phase space. Finally, we describe all two-dimension non-abelian Kähler hom-Lie algebra and also it is shown that there does not exist a non-abelian Kähler proper hom-Lie algebra of dimension two.

Keywords

• Proper hom-Lie algebra,almost Hermitian structure,hom-Levi-Civita product,Kähler hom-Lie algebra,phase space

References

1. [1] A. Andrada, M.L. Barberis, and I. Dotti, Classification of abelian complex structures on 6-dimensional Lie algebras, J. London Math. Soc. 83, 232–255, 2011.
2. [2] A. Andrada and S. Salamon, Complex product structures on Lie algebras, Forum Math. 17, 261–295, 2005.
3. [3] S. Benayadi and A. Makhlouf, Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms, J. Math. Phys. 76, 38–60, 2014.
4. [4] G. Calvaruso and A. Fino, Five-dimensional K-contact Lie algebras, Monatsh. Math. 167 (1), 35–59, 2012.
5. [5] J. Hartwig, D. Larsson, and S. Silvestrov, Deformations of Lie algebras using σ- derivations, J. Algebra 295, 314–361, 2006.
6. [6] N. Hu, q-Witt algebras, q-Lie algebras, q-holomorph structure and representations, Algebra Colloq., 6 (1), 51–70, 1999.
7. [7] J. Jiang, S.K. Mishra, and Y. Sheng, Hom-Lie algebras and hom-Lie groups, integration and differentiation, arXiv:1904.06515.
8. [8] D. Larsson and S. Silvestrov, Quasi-hom-Lie algebras, central extensions and 2- cocycle-like identities, J. Algebra 288, 321–344, 2005.

9. [9] A. Makhlouf and S.D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2), 51–64, 2008.
10. [10] A. Makhlouf and S D. Silvestrov, Notes on formal deformations of hom-associative and hom-Lie algebras, Forum Math. 22, 715–759, 2010.
11. [11] A. Makhlouf and D. Yau, Rota-baxter hom-lie admissible algebras, Comm. Algebra. 42, 1231–1257, 2014.
12. [12] E. Peyghan and L. Nourmohammadifar, Para-Kähler hom-Lie algebras, J. Algebra Appl. 18 (3), 1950044, 2019.
13. [13] E. Peyghan and L. Nourmohammadifar, Para-Kähler hom-Lie algebras of dimension 2, submitted.
14. [14] Y. Sheng and C. Bai, A new approach to hom-Lie bialgebras, J. Algebra 399, 232–250, 2014.
15. [15] Y. Sheng and D. Chen, Hom-Lie 2-algebras, J. Algebra 376, 174–195, 2013.
16. [16] Z. Xiong, Hom-Lie groups of a class of hom-Lie algebra, arXiv:1810.07881.