Jordan superalgebras with a solvable even part
Let \(G= G_0 \oplus G_1\) be a Grassmann algebra over an associative commutative ring \(\Phi\). A \(\mathbb{Z}_2\)-graded \(\Phi\)-algebra \(J= J_0 \oplus J_1\) is said to be a Jordan superalgebra if the algebra \(G(J)= J_0 \otimes G_0 \otimes J_1 \otimes G_1\) is Jordan. The main result of the paper is the following criteria for solvability of Jordan superalgebras. Theorem A. Let \(J= J_0 \oplus J_1\) be a Jordan superalgebra over \(\Phi\), and \(1/6\in \Phi\). Then \(J\) is solvable if and only if the Jordan algebra \(J_0\) is solvable. First, the author proves an analogous statement for \(\mathbb{Z}_2\)-graded Jordan algebras \(J\). This proof is based upon clever computations in the multiplication algebra \(R(J)\). Then, Theorem A follows. As a consequence of this theorem the author obtains a criteria for solvability of color Jordan superalgebras.
scientific article; zbMATH DE number 862294