Kähler-Norden structures on Hom-Lie groups and Hom-Lie algebras

Year 2025, , 213 - 236, 28.02.2025

E. Peyghan , Leila Nourmohammadifar , Abdenacer Makhlouf , Aydın Gezer

https://doi.org/10.15672/hujms.1344687

Abstract

The aim of this paper is to describe two geometric notions, holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups, and study their relationships in the left invariant setting. We study Kähler-Norden structures with abelian complex structures and give the curvature properties of holomorphic Norden structures on Hom-Lie groups. Finally, we show that any left-invariant holomorphic Hom-Lie group is a flat (holomorphic Norden Hom-Lie algebra carries a Hom-Left-symmetric algebra) if its left-invariant complex structure (complex structure) is abelian.

Keywords

Hom-Lie group, Hom-Lie algebra, holomorphic Norden structure, Kähler-Norden structure

References

• [1] A. Borowiec, M. Ferraris, M. Francaviglia and I. Volovich, Almost-complex and almost-product Einstein manifolds from a variational principle, J. Math. Phys. 40 (7), 3446–3464, 1999.
• [2] A. Borowiec, M. Francaviglia and I. Volovich, Anti-Kählerian manifolds, Differential Geom. Appl. 12 (3), 281–289, 2000.
• [3] L. Cai, J. Liu and Y. Sheng, Hom-Lie algebroids, Hom-Lie bialgebroids and Hom- Courant algebroids, J. Geom. Phys. 121, 15–32, 2017.
• [4] N. Degirmenci and S. Karapazar, Spinors on Kähler-Norden manifolds, J. Nonlinear Math. Phys. 17 (1), 27–34, 2010.
• [5] N. Degirmenci and S. Karapazar, Schrödinger-Lichnerowicz like formula on Kähler- Norden manifolds, Int. J. Geom. Meth. Mod. Phys. 9 (1), 1250010, 14 pp., 2012.
• [6] E. A. Fernández-Culma and Y. Godoy, Anti-Kählerian geometry on Lie groups, Math. Phys. Anal. Geom. 21 (8), 1–24, 2018.
• [7] G. T. Ganchev and A. V. Borisov, Note on the almost complex manifolds with Norden metric, Compt. Rend. Acad. Bulg. Sci. 39 (5), 31–34, 1986.
• [8] K. I. Gribachev, D. G. Mekerov and G. D. Djelepov, Generalized B-manifold, Compt. Rend. Acad. Bulg. Sci. 38 (3), 299–302, 1985.
• [9] J. Hartwig, D. Larsson and S. Silvestrov, Deformations of Lie algebras using $\sigma$- derivations, J. Algebra 295, 314–361, 2006.
• [10] N. Hu, q-Witt algebras, q-Lie algebras, q-holomorph structure and representations, Algebra Colloq. 6 (1), 51-70, 1999.
• [11] M. Iscan and A. A. Salimov, On Kähler-Norden manifolds, Proc. Math. Sci. 119 (1), 71–80, 2009.
• [12] J. Jiang, S. K. Mishra and Y. Sheng, Hom-Lie algebras and Hom-Lie groups, integration and differentiation, SIGMA Symmetry Integrability Geom. Methods Appl. 16, Paper No. 137, 22 pp., 2020.
• [13] D. Larsson and S. Silvestrov, Quasi-Hom-Lie algebras, central extensions and 2- cocycle-like identities, J. Algebra 288, 321–344, 2005.
• [14] C. Laurent-Gengoux, A. Makhlouf and J. Teles, Universal algebra of a Hom-Lie algebra and group-like elements, J. Pure Appl. Algebra 222 (5), 1139–1163, 2018.
• [15] A. Makhlouf and S. D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2, 51–64, 2008.
• [16] A. Makhlouf and S. D. Silvestrov, Notes on formal deformations of Hom-associative and Hom-Lie algebras, Forum Math. 22, 715–759, 2010.
• [17] A. Makhlouf and D. Yau, Rota-Baxter Hom-Lie admissible algebras, Commun. Algebra 42, 1231–1257,2014.
• [18] A. Nannicini, Generalized geometry of Norden manifolds, J. Geom. Phys. 99, 244–255, 2016.
• [19] L. Nourmohammadifar and E. Peyghan, Complex product structures on Hom-Lie algebras, Glasgow Math. J. 61, 69–84, 2019.
• [20] K. Olszak, On the Bochner conformal curvature of Kähler-Norden manifolds, Cent. Eur. J. Math. 3 (2), 309–317, 2005.
• [21] E. Peyghan and L. Nourmohammadifar, Para-Kähler Hom-Lie algebras, J. Algebra Appl. 18 (3), 1950044, 24 pp., 2019
• [22] E. Peyghan and L. Nourmohammadifar, Complex and Kähler structures on Hom-Lie algebras, Hacet. J. Math. Stat. 49 (3), 10391056, 2020.
• [23] E. Peyghan and L. Nourmohammadifar, Hom-left symmetric algebroids , Asian-Eur. J. Math. 11 (2), 1850027, 24 pp., 2018.
• [24] E. Peyghan, L. Nourmohammadifar and I. Mihai, Para-Sasakian geometry on Hom- Lie groups, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 115 (4), Paper No. 163, 22 pp., 2021.
• [25] Y. Sheng and C. Bai, A new approach to Hom-Lie bialgebras, J. Algebra 399, 232–250, 2014.
• [26] K. Słuka, On Kähler manifolds with Norden metrics, An. tiin. Univ. Al. I. Cuza Iai. Mat. (N.S.) 47, 105–122, 2001.
• [27] K. Słuka, On the curvature of Kähler-Norden manifolds, J. Geom. Phys. 54 (2), 131–145, 2005.
• [28] D. Yau, Hom-Novikov algebras, J. Phys. A 44, 085202, 20 pp., 2011.