Ricerc@Sapienza

It has been shown that in characteristic zero the generators of the minimal supervarieties of finite basic rank belong to the class of minimal superalgebras introduced by Giambruno and Zaicev (Trans Am Math Soc 355:5091-5117, 2003). In the present paper the complete list of minimal supervarieties generated by minimal superalgebras whose maximal semisimple homogeneous subalgebra is the sum of three graded simple algebras is provided. As a consequence, we negatively answer the question of whether any minimal superalgebra generates a minimal supervariety.

Let $F$ be a field of characteristic zero and $p$ a prime. In the present paper it is proved that a variety of $Z_p$-graded associative PI $F$-algebras of finite basic rank is minimal of fixed $Z_p$-exponent $d$ if, and only if, it is generated by an upper block triangular matrix algebra $UT_{Z_p}(A_1,ldots, A_m)$ equipped with a suitable elementary $Z_p$-grading, whose diagonal blocks are isomorphic to $Z_p$-graded simple algebras $A_1,ldots,A_m$ satisfying $dim_F (A_1 opluscdotsoplus A_m)=d$.