Minimal supervarieties with factorable ideal of graded polynomial identities

Minimal varieties of graded PI‐algebras over abelian groups

“…In [16], the authors proved that every minimal affine variety of $$PI$$‐superalgebras is generated by one of the minimal superalgebras introduced by Giambruno and Zaicev in the ordinary case. Despite some partial and positive results in [11], not every minimal superalgebra generates a minimal variety (see [12]), so we need a more refined list of generating algebras to achieve the aim of classification, even in the case of affine varieties.…”

A characterization of minimal varieties of Zp-graded PI algebras

“…In particular, it has been established that any such supervariety is generated by one of the above mentioned minimal superalgebras. But despite some partial results in [8] and [9], until this moment their complete characterization is still unknown.…”

Minimal superalgebras generating minimal supervarieties

“…Also A ss can be written as the direct sum of graded simple algebras which can be of two types: either simple or non-simple as algebras. It has been proved that in the case in which the sequence of the graded simple components of A ss has in some sense a regular distribution, the supervariety generated by A is minimal (Theorems 4.7 and 5.4 of [4] and 3.6 of [3]).…”

“…Let us consider the superalgebras isomorphisms Ψ jj : A j −→ B j such that Ψ jj (e j ) = f j if j = 2 and Ψ 22 (ρ 2 ) = ν 2 (and hence Ψ 22 (ρ 2 ) = ν2 ). Applying the same arguments of the previous Section, for every 1 ≤ i < j ≤ 3 one constructs a vector space isomorphism Ψ ij from the subspace A ij of A into the subspace B ij of B, which clearly preserves the Z 2 -grading when (i, j) = (1,3). For what concerns the latter case, for the map…”

“…. But, according to the discussion of Section 2 of [3], in any event W is a minimal superalgebra with maximal semisimple homogeneous subalgebra coinciding with A 2 ⊕ A 3 . At this stage, from Theorem 5.3 of [4] one has that T Z 2 (W ) = T Z 2 (A 2 ) • T Z 2 (A 3 ), and this concludes the proof.…”