×

Growth of some varieties of Leibniz-Poisson algebras. (English) Zbl 1265.17024

An algebra \(A(+,\cdot,\{,\})\) over a field \(K\) with two binary multiplications \(\cdot\) and \(\{\,,\,\}\) is called Leibniz-Poisson if it is a unital commutative-associative algebra with respect to \(\cdot\), a Leibniz algebra with respect to \(\{\,,\,\}\), and these two operations are connected by the relations \(\{a\cdot b,c\}=a\cdot\{b,c\}+\{a,c\}\cdot b\), \(\{c,a\cdot b\}=a\cdot\{c,b\}+\{c,a\}\cdot b\), \(a,b,c\in A\). In the paper under review the author studies varieties \(\mathcal V\) of Leibniz-Poisson algebras over an arbitrary field \(K\) whose ideal of identities contains the identities \[ \{\{x_1,y_1\},\{x_2,y_2\},\ldots,\{x_m,y_m\}\}=0,~~ \{x_1,y_1\}\cdot\{x_2,y_2\}\cdot ... \cdot\{x_m,y_m\}=0 \] for some \(m\). The first main result gives that there exist constants \(\alpha,\beta\) and a positive integer \(d\) such that the codimension sequence \(c_n({\mathcal V})\) satisfies the inequalities \(n^{\alpha}d^n\leq c_n({\mathcal V})\leq n^{\beta}d^n\) for sufficiently large \(n\). Hence the exponent \(\text{ Exp}({\mathcal V})\) of \(\mathcal V\) exists and is a positive integer. The second main result concerns the cocharacter sequence \(\chi_n({\mathcal V})=\sum_{\lambda\vdash n}m_{\lambda}\chi_{\lambda}\), \(\lambda=(\lambda_1,\ldots,\lambda_n)\), of the varieties \(\mathcal V\) in consideration when \(\text{ char}(K)=0\). It is shown that the fact \(\text{ Exp}({\mathcal V})\leq d\) is equivalent to the condition that there is a constant \(C\) such that \(m_{\lambda}=0\) if \(\lambda_{d+1}+\cdots+\lambda_n>C\).

MSC:

17B63 Poisson algebras
17A32 Leibniz algebras
17A30 Nonassociative algebras satisfying other identities
PDFBibTeX XMLCite