Graded identities for algebras with elementary gradings over an infinite field

Let $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$ -grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$ , $b\in \mathbb {N}$ , we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$ , as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$ -algebras which are not PI equivalent. Actually, we prove that the $T_{G}$ -ideal of the former algebra is contained in the $T$ -ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

Some graded identities of the Cayley–Dickson algebra

We work to find a basis of graded identities for the octonion algebra. We do so for the [Formula: see text] and [Formula: see text] gradings, both of them derived of the Cayley–Dickson (C–D) process, the later grading being possible only when the characteristic of the scalars is not two.

ℤ2-graded codimensions of unital algebras

We study polynomial identities of nonassociative algebras constructed by using infinite binary words and their combinatorial properties. Infinite periodic and Sturmian words were first applied for constructing examples of algebras with an arbitrary real PI-exponent greater than one. Later, we used these algebras for a confirmation of the conjecture that PI-exponent increases precisely by one after adjoining an external unit to a given algebra. Here, we prove the same result for these algebras for graded identities and graded PI-exponent, provided that the grading group is cyclic of order two.