Semisimple algebras and PI-invariants of finite dimensional algebras

Let $\Gamma$ be the $T$-ideal of identities of an affine PI-algebra over an algebraically closed field $F$ of characteristic zero. Consider the family ${\mathcal{ℳ}}_{\Gamma }$ of finite dimensional algebras $\mathrm{\Sigma }$ with $\mathrm{Id}<!--FUNCTION\; APPLICATION-->(\mathrm{\Sigma })=\Gamma$. By Kemer’s theory ${\mathcal{ℳ}}_{\Gamma }$ is not empty. We show there exists $A\in {\mathcal{ℳ}}_{\Gamma }$ with Wedderburn–Malcev decomposition $A\cong <!--FUNCTION\; APPLICATION-->A_{ss}\oplus J_A$, where $J_A$ is the Jacobson’s radical and $A_{ss}$ is a semisimple supplement with the property that if $B\cong <!--FUNCTION\; APPLICATION-->B_{ss}\oplus J_B\in {\mathcal{ℳ}}_{\Gamma }$ then $A_{ss}$ is a direct summand of $B_{ss}$. In particular $A_{ss}$ is unique minimal, thus an invariant of $\Gamma$. More generally, let $\Gamma$ be the $T$-ideal of identities of a PI algebra and let ${\mathcal{ℳ}}_{{\mathbb{Z}}_2,\Gamma }$ be the family of finite dimensional superalgebras $\mathrm{\Sigma }$ with $\mathrm{Id}<!--FUNCTION\; APPLICATION-->(E(\mathrm{\Sigma }))=\Gamma$. Here $E$ is the unital infinite dimensional Grassmann algebra and $E(\mathrm{\Sigma })$ is the Grassmann envelope of $\mathrm{\Sigma }$. Again, by Kemer’s theory ${\mathcal{ℳ}}_{{\mathbb{Z}}_2,\Gamma }$ is not empty. We prove there exists a superalgebra $A\cong <!--FUNCTION\; APPLICATION-->A_{ss}\oplus J_A\in {\mathcal{ℳ}}_{{\mathbb{Z}}_2,\Gamma }$ such that if $B\in {\mathcal{ℳ}}_{{\mathbb{Z}}_2,\Gamma }$, then $A_{ss}$ is a direct summand of $B_{ss}$ as superalgebras. Finally, we fully extend these results to the $G$-graded setting where $G$ is a finite group. In particular we show that if $A$ and $B$ are finite dimensional $G_2:={\mathbb{Z}}_2\times G$-graded simple algebras then they are $G_2$-graded isomorphic if and only if $E(A)$ and $E(B)$ are $G$-graded PI-equivalent.