Graded Identities and Central Polynomials for the Verbally Prime Algebras

Graded identities for algebras with elementary gradings over an infinite field

“…Variants of the following lemma were proved in various situations, to the best of our knowledge, a similar statement first appeared in [30, Lemma 6]. In our case, the proof of the lemma follows word by word the one of [20,Lemma 6.4], and that is why we omit it here.…”

“…As a consequence, we exhibit a basis for the graded identities of the algebras M a,b (E) with respect to these elementary gradings, as well as their tensor products, once again up to graded monomial identities. The problem of describing these monomial identities is still open even in characteristic zero, and it is still far from being understood, although it was done in several particular cases, see [20].…”

“…Let (g, a) ∈ G * = G × Z 2 and define the following matrices in M mn (R Over a field of characteristic zero, it was proved in [20,Theorem 6.11] that the sets T G * (M ar+bs,as+br (E)) and T G * (M a,b (E) ⊗ M r,s (E)) coincide. Such a condition may fail when the field is infinite of positive characteristic p. In [3], whenever p > 2, the authors constructed an ordinary polynomial identity for M a,b (E) ⊗ M r,s (E) which is not one for M ar+bs,as+br (E), in the case (r, s) = (1, 1).…”

“…By Theorems 5.3 and 5.6, it is enough to prove that all graded monomial identities for A are graded identities for A as well. By [20,Theorem 6.11], we can consider monomials with at least one variable which appears at least twice. Let m = m(x 1 , .…”

“…Here u stands for u (mod a + b). Moreover, in [20], over a field of characteristic zero, it was provided a basis of the graded identities for M a,b (E) when equipped with the grading given in the above definition. Lemma 5.2.…”