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Discrete differential-geometric Poisson brackets: general theory and explicit constructions

  • Marta Dell'Atti
  • Giorgio Gubbiotti
  • Pierandrea Vergallo
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Abstract

We review the discrete theory of differential-geometric Poisson brackets as introduced by B. A. Dubrovin, and we present a complete proof of their characterisation in the non-degenerate case. We use this characterisation to show several novel examples of such structures, providing a complete classification for non-degenerate four-dimensional differential-geometric Poisson brackets under the additional assumption of the Lie algebra to be of quasi-Frobenius type. We also present a complete classification of quasi-Frobenius filiform N-graded Lie algebras, including two families in arbitrary dimensions.

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Cite

BibTeXRIS
BibTeX
@misc{dellatti2026discrete,
  title = {{Discrete differential-geometric Poisson brackets: general theory and explicit constructions}},
  author = {Marta Dell'Atti and Giorgio Gubbiotti and Pierandrea Vergallo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40049},
  eprint = {2609.40049},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40049},
}
RIS
TY  - UNPB
TI  - Discrete differential-geometric Poisson brackets: general theory and explicit constructions
AU  - Marta Dell'Atti
AU  - Giorgio Gubbiotti
AU  - Pierandrea Vergallo
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40049
UR  - https://arxiv.org/abs/2609.40049
AB  - We review the discrete theory of differential-geometric Poisson brackets as introduced by B. A. Dubrovin, and we present a complete proof of their characterisation in the non-degenerate case. We use this characterisation to show several novel examples of such structures, providing a complete classification for non-degenerate four-dimensional differential-geometric Poisson brackets under the additional assumption of the Lie algebra to be of quasi-Frobenius type. We also present a complete classification of quasi-Frobenius filiform N-graded Lie algebras, including two families in arbitrary dimensions.
ER  -