Superpolynomial identities of finite-dimensional simple algebras

We investigate the Grassmann envelope (of finite rank) of a finite-dimensional $\mathbb{Z}_2$-graded algebra. As a result, we describe the polynomial identities of $G_1(\mathcal{A})$, where $G_1$ stands for the Grassmann algebra with $1$ generator, and $\mathcal{A}$ is a $\mathbb{Z}_2$-graded-simple associative algebra. We also classify the conditions under which two associative $\mathbb{Z}_2$-graded-simple algebras share the same set of superpolynomial identities, i.e., the polynomial identities of its Grassmann envelope (in particular, of finite rank). Moreover, we extend the construction of the Grassmann envelope for the context of $\Omega$-algebras and prove some of its properties. Lastly, we give a description of $\mathbb{Z}_2$-graded-simple $\Omega$-algebras.