A characterization of minimal varieties of Zp-graded PI algebras

Minimal varieties of graded PI‐algebras over abelian groups

“…Similar results hold for simple Lie algebras [27], simple Jordan algebras [17], simple nonassociative algebras [30], and, finally, this was proved in the very general setting of finite‐dimensional simple $$\Omega$$‐algebras [6]. The same problem has been tackled also for $$G$$‐graded algebras of upper block triangular matrices [5, 13, 14]. Here, we consider the case where the diagonal blocks are simple graded and $$G$$ is a finite abelian group.…”

“…The importance of minimal graded algebras is given by the next results stated in [13] for an arbitrary finite group $$G$$. Lemma Let $$A$$ be a finite‐dimensional $$G$$‐graded algebra over an algebraically closed field.…”

“…This list has been fully described in the case of algebras graded by groups of order a prime number. More precisely, if $$p$$ is a prime, in [13], the authors proved that an affine variety of $$\mathbb {Z}_p$$‐graded PI‐algebras is minimal if and only if is generated by an algebra of upper triangular block matrices $$UT(A_1,\dots,A_m)$$ built on some sequence of finite‐dimensional $$\mathbb {Z}_p$$‐graded simple algebras $$A_1,\dots,A_m$$.…”

Gradings on the algebra of triangular matrices as a Lie algebra: revisited

“…We recall that block-triangular matrices play an important role in PI-theory, see, for instance, [20]. Their related graded PI-properties have been a subject of several recent studies, see [10,16,15,5,13,14,17], to cite only a few examples. The group gradings on the upper block-triangular matrices were computed in [33,6,35].…”