The algebraic classification and degenerations of nilpotent Poisson algebras (Q2099259)

A Poisson algebra \(\mathcal{P}\) is a commutative algebra over a field equipped with a Lie bracket \(\{-,-\}:\mathcal{P} \times \mathcal{P} \to \mathcal{P}\) which is a biderivation on \(\mathcal{P}\). The authors give a definition of nilpotence for Poisson ideals and Poisson algebras. The primary results are a classification of Poisson algebras of dimension at most 4. On the algebraic side, their classification follows the work of \textit{T. Skjelbred} and \textit{T. Sund} [C. R. Acad. Sci., Paris, Sér. A 286, 241--242 (1978; Zbl 0375.17006)] for nilpotent Lie algebras. The authors offer an alternative, geometric approach to their classification. This involves recognizing the subclass of \(n\)-dimensional nilpotent Poisson algebras as a subvariety of \(n\)-dimensional Poisson algebras. They relate the classification problem to classifying the irreducible components of the variety.