\((n+1)\)-ary derivations of simple \(n\)-ary algebras
In this short paper the author states that the following two theorems hold true. Theorem 1. A simple binary Mal'tsev algebra \(M_7\) (a ternary Mal'tsev algebra \(M_8\)) does not have nontrivial 3-ary (resp. 4-ary) derivations. Theorem 2. Let \((f_0, f_1, \ldots, f_n)\) be an \((n+1)\)-ary derivation of a simple finite-dimensional \(n\)-ary Filippov algebra over an algebraically closed field of characteristic zero. Then \[ (f_0, f_1, \ldots, f_n)=\left(\sum_{j=1}^n h_j\cdot \text{id}, h_1\cdot \text{id},\ldots,h_n\cdot \text{id}\right)+(d_0,d,\ldots,d) \] and \(\frac{\text{tr}([d_0])}n E-[d_0]^T=[d]\), where \([d_0]\) and \([d]\) are matrices of linear transformations \(d_0\) and \(d\)
scientific article; zbMATH DE number 6115019