Skip to content
DraconiticResearch
Search Sources API
181,686 studies31 sourcesUpdated 46 min ago
PreprintCC BY 4.0arXiv

Classification of Hom--Lie structures on finite-dimensional simple Lie algebras

  • Jiacheng Sun
  • Chi Zhang
  • Haoran Zhu
Open at sourcePDFDOICite

Abstract

We classify the Hom--Lie structures on finite-dimensional simple Lie algebras over an algebraically closed field of characteristic different from 2 and 3. Except for the three-dimensional simple Lie algebra and the Zassenhaus algebras, every such structure is a scalar multiple of the identity. In the two exceptional cases, we determine the full Hom--Lie space: it has dimension six for the three-dimensional algebra, while for a Zassenhaus algebra it consists precisely of the multiplication operators arising from the corresponding divided-power algebra. This proves a conjecture of Makhlouf and Zusmanovich.

Related

  1. Classification of Finite-Dimensional Simple Novikov Algebras and Their Irreducible Bimodules in Positive Characteristic
    Dong Liu, Yufeng Pei, Limeng Xia
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryMathematical physicsarXiv
  2. Transferred L∞-structures and a classification of binary brackets for finite-dimensional skew-gentle algebras
    Zixu Li, Liyu Liu, Lingchao Meng
    arXivPreprintCC0Full textAlgebra & number theoryarXiv
  3. Classification of Nijenhuis operators on three-dimensional multiplicative simple Bihom-Lie algebras
    Bouzid Mosbahi
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  4. Idempotent words in finite-dimensional simple Jordan algebras
    Ilja Gogić, Mateo Tomašević
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories
    Bowen Li
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{sun2026classification,
  title = {{Classification of Hom--Lie structures on finite-dimensional simple Lie algebras}},
  author = {Jiacheng Sun and Chi Zhang and Haoran Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25056},
  eprint = {2610.25056},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25056},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Classification of Hom--Lie structures on finite-dimensional simple Lie algebras
AU  - Jiacheng Sun
AU  - Chi Zhang
AU  - Haoran Zhu
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25056
UR  - https://arxiv.org/abs/2610.25056
AB  - We classify the Hom--Lie structures on finite-dimensional simple Lie algebras over an algebraically closed field of characteristic different from 2 and 3. Except for the three-dimensional simple Lie algebra and the Zassenhaus algebras, every such structure is a scalar multiple of the identity. In the two exceptional cases, we determine the full Hom--Lie space: it has dimension six for the three-dimensional algebra, while for a Zassenhaus algebra it consists precisely of the multiplication operators arising from the corresponding divided-power algebra. This proves a conjecture of Makhlouf and Zusmanovich.
ER  -