I'm sorry if the question is stupid.
I'm currently studying the $\mathbb Z$-graded algebras, for example, the cyclotomic quiver Hecke algebras, and I'm starting to wonder why we want to know the grading of an algebra.
I have searched the internet for a while but nothing too useful comes out. Can anyone give me some motivations for finding the grading of the algebras, or at least, the motivation of showing that the cyclotomic Hecke algebras of type A is graded please? Many many thanks.
A grading by a group $\Gamma$ on an algebra $A$ is an additional structure, which often is very useful and appears in a number of contexts, including representations of finite groups, Lie theory, representation theory and geometry. For example, the local classification of symmetric Riemannian spaces reduces to the classification of $\mathbb{Z}_2$-graded complex simple Lie algebras.
As to the cyclotomic quiver Hecke algebra, it is $\mathbb{Z}$-graded by definition, see section $2$ (in particular definition $2.2.9$) in the article Cyclotomic quiver Hecke algebra of type A. As to the motivation for studying these $\mathbb{Z}$-graded algebras, the article mentions $2$-representation theory and questions from geometry.