PreprintCC BY 4.0arXiv
Dedekind Poisson Algebras over Arbitrary Fields
Abstract
We study Poisson algebras in which every Poisson subalgebra is a Poisson ideal, called Dedekind Poisson algebras, over arbitrary fields and without any finite-dimensionality assumption. In characteristic different from 2, combining the two operations by x*y=xy+[x,y] connects the problem with Outcalt's classical classification of power-associative H-algebras and yields the same structural type. We obtain a complete classification by a direct two-operation argument valid in every characteristic. This approach also shows that characteristic 2 retains additional Poisson information which cannot be recovered from the single product *.
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BibTeX
@misc{plakosh2026dedekind,
title = {{Dedekind Poisson Algebras over Arbitrary Fields}},
author = {A. I. Plakosh and O. O. Pypka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13767},
eprint = {2609.13767},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13767},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dedekind Poisson Algebras over Arbitrary Fields
AU - A. I. Plakosh
AU - O. O. Pypka
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13767
UR - https://arxiv.org/abs/2609.13767
AB - We study Poisson algebras in which every Poisson subalgebra is a Poisson ideal, called Dedekind Poisson algebras, over arbitrary fields and without any finite-dimensionality assumption. In characteristic different from 2, combining the two operations by x*y=xy+[x,y] connects the problem with Outcalt's classical classification of power-associative H-algebras and yields the same structural type. We obtain a complete classification by a direct two-operation argument valid in every characteristic. This approach also shows that characteristic 2 retains additional Poisson information which cannot be recovered from the single product *.
ER -