For a Lie algebra $L$, let $S(L)$ denote the symmetric Poisson algebra and $\mathbf{s}(L)$ the truncated symmetric Poisson algebra of $L$. In characteristic $p\ne 2$, the conditions under which these Poisson algebras are solvable were established by Monteiro Alves and Petrogradsky in (J. Algebra 488 (2017) 244–281). The characterization in the harder case $p=2$ was left as an open problem and a related conjecture formulated. In this paper, we prove a corrected version of that conjecture, which thereby completes the classification.

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关于对称泊松代数可解性的一个猜想

对于李代数 $升$， 让 $秒(升)$ 表示对称泊松代数和 $\mathbf{秒}(升)$ 截断的对称泊松代数 $升$. 在特点$磷\ne 2$，这些泊松代数可解的条件由 Monteiro Alves 和 Petrogradsky 在 ( J. Algebra 488 (2017) 244–281) 中建立。较难情况下的表征$磷=2$被留下作为一个悬而未决的问题，并制定了相关的猜想。在本文中，我们证明了该猜想的更正版本，从而完成了分类。