Proper \(T\)-ideals of Poisson algebras with extreme properties (Q2364184)

Let \(A\) be an associative algebra over a field \(K\) of zero characteristic. The author introduces the structure of a Poisson algebra on the vector space \(A\oplus A\oplus K\). Let \(\Lambda_{2k}\) be the Grassmann algebra in \(2k\) variables and \(N_k\) a subalgebra of upper triangular matrices of size \(k\) introduced in [\textit{A. Giambruno} et al., Isr. J. Math. 158, 367--378 (2007; Zbl 1127.16018)]. The author considers two varieties of Poisson algebras generated by algebras \(\Lambda_{2k}\oplus \Lambda_{2k}\oplus K\) and \(N_k\oplus N_k\oplus K\). They have the following extremal property. Their so called \textit{proper codimension sequences} \(\{\gamma(n)\mid n\geq 1\}\) have a polynomial growth, but the proper codimension sequences for their proper subvarieties are polynomial of strictly smaller degrees. These two varieties of Poisson algebras are studied in more details.