Abstract
We generalize the Skjelbred–Sund method, used to classify nilpotent low-dimensional Lie algebras, in order to classify Poisson algebras with non-trivial annihilator. We develop this method with the purpose of classifying nilpotent Poisson algebras, obtaining from it the algebraic classification of the nilpotent Poisson algebras up to dimension four. Additionally, we obtain the geometric classification of the variety of nilpotent Poisson algebras up to dimension four, by adapting some notions and results used for varieties of algebras with a single multiplication.
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