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PreprintCC BY 4.0arXiv

Kählerian oxidation and reduction of Lie algebras with Kähler structures

  • Mustapha Bachaou
  • Ignacio Bajo
  • Mohamed Louzari
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Abstract

A Kähler structure on a real Lie algebra g is a pair $(ω, J)$ of a 2-cocycle and a complex structure on g which are compatible in the sense that J is skew-symmetric with respect to ω. This paper investigates Kähler structures on Lie algebras, with a particular focus on the nilpotent case. We introduce an algebraic framework for the Kählerian reduction and oxidation of Lie algebras by one-dimensional and two-dimensional ideals. Our main structural result demonstrates that every indecomposable, quasi-nilpotent Kählerian nilpotent Lie algebra of dimension greater than four can be inductively reconstructed through a finite sequence of Kählerian central oxidations by planes. This reduction sequence originates from a base algebra that is either {0}, ℝ2, or one admitting a strongly non-nilpotent complex structure. Moreover, if the complex structure is nilpotent, all intermediate reductions also have a nilpotent complex structure.

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Cite

BibTeXRIS
BibTeX
@misc{bachaou2026kählerian,
  title = {{Kählerian oxidation and reduction of Lie algebras with Kähler structures}},
  author = {Mustapha Bachaou and Ignacio Bajo and Mohamed Louzari},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26417},
  eprint = {2609.26417},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26417},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Kählerian oxidation and reduction of Lie algebras with Kähler structures
AU  - Mustapha Bachaou
AU  - Ignacio Bajo
AU  - Mohamed Louzari
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26417
UR  - https://arxiv.org/abs/2609.26417
AB  - A Kähler structure on a real Lie algebra g is a pair $(ω, J)$ of a 2-cocycle and a complex structure on g which are compatible in the sense that J is skew-symmetric with respect to ω. This paper investigates Kähler structures on Lie algebras, with a particular focus on the nilpotent case. We introduce an algebraic framework for the Kählerian reduction and oxidation of Lie algebras by one-dimensional and two-dimensional ideals. Our main structural result demonstrates that every indecomposable, quasi-nilpotent Kählerian nilpotent Lie algebra of dimension greater than four can be inductively reconstructed through a finite sequence of Kählerian central oxidations by planes. This reduction sequence originates from a base algebra that is either {0}, ℝ^2, or one admitting a strongly non-nilpotent complex structure. Moreover, if the complex structure is nilpotent, all intermediate reductions also have a nilpotent complex structure.
ER  -