Minimal superalgebras generating minimal supervarieties

Abstract

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 355:5091â\u80\u935117, 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

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.…”

“…In particular, the combination of the last mentioned result with what we proved when A k and B k are non-simple graded simple guarantees for each A simply by H (k) . In order to prove that A and B are isomorphic as G-graded algebras, it is enough to show that there exists g ∈ G such that (9) w…”

“…m], the equality ( 9) is satisfied for any choice of g ∈ G. On the other hand, if there exists a unique integer ∈ [1, m] such that H ( ) = 1 G , it is sufficient to set g := g to obtain (9). Therefore we are left with the case in which there exist at least two…”