Minimal varieties of PI-superalgebras with graded involution

Abstract

In the present paper it is proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed * -graded exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary Z2-grading and graded involution.

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

“…, A m ), of a suitable upper block triangular matrix algebra endowed with an elementary grading preserved by the flip along the secondary diagonal, in which, roughly speaking, each *superalgebra A i is embedded at the i-th block of the main diagonal. The main result they stated is the following Theorem 2.1 (2.2 of [7]). A variety of PI * -superalgebras of finite basic rank is minimal of * -graded exponent d if, and only if, it is generated by a…”

“…In [7] the authors classified minimal varieties of PI * -superalgebras of finite basic rank, that is, generated by a finitely generated PI * -superalgebra, of given * -graded exponent. We recall that given a…”

“…. , A m ), as constructed in [7], starting from that of the simple components A i . For this, it is necessary to recall the unique (up to graded isomorphisms) finite-dimensional simple superalgebras, namely:…”

“…, A m ) be an m-tuple of finite-dimensional simple *superalgebras. As observed in [7], it is possible to realize each of them as a subalgebra of an upper block triangular matrix algebra. To see this, let γ n be the orthogonal involution on M n , namely the map sending each a ∈ M n into the element a γn ∈ M n obtained reflecting a along its secondary diagonal (that is, for any matrix unit…”

“…Again in the case of finite basic rank, Di Vincenzo and Spinelli in [8] classified minimal varieties of algebras with involution (even if there they dealt with varieties of finite-dimensional * -algebras, according to [2] the result can be extended to varieties generated by finitely generated PI * -algebras) and, very recently, jointly with da Silva made the same for varieties of Z p -graded PI-algebras and PI * -superalgebras in [6] and [7], respectively. Roughly speaking, for each of these classes of algebras, given a sequence of simple algebras (A 1 , .…”