On the factorability of the ideal of ⁎-graded polynomial identities of minimal varieties of PI ⁎-superalgebras

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 Ω-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.…”

“…Furthermore, 𝐴 𝑠𝑠 can be written as the direct sum of graded simple algebras (or 𝐺-simple algebras), that is, graded algebras without nontrivial graded ideals. Definition 2.2 (Definition 2.2 of [13]). Let 𝐹 be an algebraically closed field.…”

“…The importance of minimal graded algebras is given by the next results stated in [13] for an arbitrary finite group 𝐺. Lemma 2.3. 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 ℤ 𝑝graded PI-algebras is minimal if and only if is generated by an algebra of upper triangular block matrices 𝑈𝑇(𝐴 1 , … , 𝐴 𝑚 ) built on some sequence of finite-dimensional ℤ 𝑝 -graded simple algebras 𝐴 1 , … , 𝐴 𝑚 .…”

Central polynomials of graded algebras: Capturing their exponential growth

“…Minimal varieties were completely described in [17], [11] and [10] in the setting of ordinary polynomial identities, identities with involution and identities with graded involution, respectively. Moreover, in [9] the authors studied minimal varieties for algebras graded by the cyclic group Z p . In what follows we present some basic results strictly related to our purpose.…”