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PreprintCC BY 4.0arXiv

Nilpotent Lie algebras of nilpotence class at most three via graphs

  • Andrea L. Gallo
  • Denis E. Videla
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Abstract

In this work, we associate with each finite simple graph G a nilpotent Lie algebra nG of class at most three, whose underlying vector space has a basis indexed by the vertices, edges, and paths of length two of G. Its nilpotency class is exactly three if and only if G contains a path of length two. We compute the dimensions of its lower central series and describe its center. Over fields of characteristic different from two, we prove that two such Lie algebras are isomorphic if and only if their underlying graphs are isomorphic. We also give an algebraic description using quotients involving exterior and symmetric squares. Finally, we study balanced blow-ups and prove that nG is a retract of nG(m).

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Cite

BibTeXRIS
BibTeX
@misc{gallo2026nilpotent,
  title = {{Nilpotent Lie algebras of nilpotence class at most three via graphs}},
  author = {Andrea L. Gallo and Denis E. Videla},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33963},
  eprint = {2610.33963},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33963},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Nilpotent Lie algebras of nilpotence class at most three via graphs
AU  - Andrea L. Gallo
AU  - Denis E. Videla
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.33963
UR  - https://arxiv.org/abs/2610.33963
AB  - In this work, we associate with each finite simple graph G a nilpotent Lie algebra n_G of class at most three, whose underlying vector space has a basis indexed by the vertices, edges, and paths of length two of G. Its nilpotency class is exactly three if and only if G contains a path of length two. We compute the dimensions of its lower central series and describe its center. Over fields of characteristic different from two, we prove that two such Lie algebras are isomorphic if and only if their underlying graphs are isomorphic. We also give an algebraic description using quotients involving exterior and symmetric squares. Finally, we study balanced blow-ups and prove that n_G is a retract of n_G^(m).
ER  -