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 $2$. % Let $\mathsf{G}$ be a finite abelian group, $\mathbb{F}$ a field of characteristic zero and $\mathfrak{A}$ a $\mathsf{G}$-graded $\mathbb{F}$-algebra. % We prove that, for $\mathbb{F}$ algebraically closed, if $\mathfrak{A}_e$ satisfies a polynomial identity $g=g(x_1^{(e)}, \dots, x_n^{(e)})\in\mathbb{F}\langle X^\mathsf{G} \rangle$ of degree $2$, then $\mathfrak{A}$ is either nilpotent or has commutative neutral component, % and we ensure that the $\mathsf{G}$-graded variety $\mathfrak{W}^\mathsf{G}$ determined by $g$ is equal to either $\mathsf{var}^\mathsf{G}([x^{(e)},y^{(e)}])$ or $\mathsf{var}^\mathsf{G}(N)$ for some nilpotent $\mathsf{G}$-graded algebra $N$. % Posteriorly, we investigate the implications of $\mathfrak{A}_e$ being central in $\mathfrak{A}$. The results obtained allow us to prove that, when $\mathsf{G}$ is finite cyclic, if $\mathfrak{A}$ is finitely generated and $\mathfrak{A}_e$ is central in $\mathfrak{A}$, then the commutator ideal of $\mathfrak{A}$ is nilpotent, and the algebra $\mathfrak{A}^{(-)}=(\mathfrak{A},[\ ,\ ])$ is a solvable Lie algebra, % and, if $\mathsf{G}$ has odd order, then $[x_1,x_2][x_3,x_4]\cdots[x_{2d-1},x_{2d}]\equiv0$ in $\mathfrak{A}$, for some $d\in\mathbb{N}$.