The Graded Algebras with a Graded Identity of Degree 2

This paper is devoted to the study of graded associative algebras that satisfy a graded polynomial identity of degree [Formula: see text]. Let [Formula: see text] be a finite abelian group, [Formula: see text] a field of characteristic zero and [Formula: see text] a [Formula: see text]-graded [Formula: see text]-algebra. We prove that, for [Formula: see text] algebraically closed, if [Formula: see text] satisfies a polynomial identity [Formula: see text] of degree [Formula: see text], then [Formula: see text] is either nilpotent or has commutative neutral component, and we show that the [Formula: see text]-graded variety [Formula: see text] determined by [Formula: see text] is equal to either [Formula: see text] or [Formula: see text] for some nilpotent [Formula: see text]-graded algebra [Formula: see text]. Posteriorly, we investigate the case where [Formula: see text] is central in [Formula: see text]. The results obtained allow us to prove that, when [Formula: see text] is finite cyclic, if [Formula: see text] is finitely generated and [Formula: see text] is central in [Formula: see text], then the commutator ideal of [Formula: see text] is nilpotent, and the algebra [Formula: see text] is a solvable Lie algebra, and, if [Formula: see text] has odd order, then [Formula: see text] in [Formula: see text], for some [Formula: see text].