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Identities for generalized Bernoulli numbers from a Lie model of the 2-disk

  • Urtzi Buijs
  • José Manuel Moreno Fernández
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Abstract

We define generalized Bernoulli numbers as appropriate coefficients of the Baker--Campbell--Hausdorff (BCH) product formula, extending the classical sequence of rational numbers to a 2-dimensional setting. We give an identity involving these numbers and the classical Bernoulli numbers which is equivalent to the existence of a particular complete differential graded Lie algebra modelling a cell decomposition of the 2-disk and based on the Lawrence--Sullivan model of the interval. From this identity we derive explicit quadratic relations. The existence of Lie models for the symmetries of the interval and the disk is related to certain properties of the BCH product and to some notable features of the classical Bernoulli numbers: the two choices for B1, the vanishing of Bn for odd n≥ 3, and the equality B4=B8.

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Cite

BibTeXRIS
BibTeX
@misc{buijs2026identities,
  title = {{Identities for generalized Bernoulli numbers from a Lie model of the 2-disk}},
  author = {Urtzi Buijs and José Manuel Moreno Fernández},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00825},
  eprint = {2611.00825},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00825},
}
RIS
TY  - UNPB
TI  - Identities for generalized Bernoulli numbers from a Lie model of the 2-disk
AU  - Urtzi Buijs
AU  - José Manuel Moreno Fernández
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00825
UR  - https://arxiv.org/abs/2611.00825
AB  - We define generalized Bernoulli numbers as appropriate coefficients of the Baker--Campbell--Hausdorff (BCH) product formula, extending the classical sequence of rational numbers to a 2-dimensional setting. We give an identity involving these numbers and the classical Bernoulli numbers which is equivalent to the existence of a particular complete differential graded Lie algebra modelling a cell decomposition of the 2-disk and based on the Lawrence--Sullivan model of the interval. From this identity we derive explicit quadratic relations. The existence of Lie models for the symmetries of the interval and the disk is related to certain properties of the BCH product and to some notable features of the classical Bernoulli numbers: the two choices for B_1, the vanishing of B_n for odd n≥ 3, and the equality B_4=B_8.
ER  -