Grassmann speciality of Jordan supersystems (Q1879652)

Let \(k\) be a basic commutative associative ring with a unit element such that \(\frac12\in k\). Denote by \(\Gamma\) the Grassmann algebra of a free \(k\)-module of a countable rank. Then \(\Gamma=\Gamma_0\oplus \Gamma_1\) is an associative superalgebra. The Grassmann envelope of a Jordan superalgebra \(A\) is a superalgebra \(G(A)=(A_0\otimes \Gamma_0)\oplus (A_1\otimes \Gamma_1)\) over the commutative polynomial ring \(\Gamma_0\). The aim of the paper is to discuss connections between specialty of \(A\) over \(k\) and of \(G(A)\) over \(\Gamma_0\) in the following cases: 1) \(A\) is a linear Jordan superalgebra; 2) \(A\) is a quadratic Jordan superalgebra; 3) \(A\) is a Jordan supertriple system.