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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.

Thomas Tong
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  • I think this is a great question. If you don't get a fully satisfactory answer here, I'd suggest posting on MathOverflow. – Hugh Thomas Sep 03 '14 at 22:52
  • @HughThomas I don't know if my answer is "fully satisfactory", but I think for MO the question is too broad and should be more specific. – Dietrich Burde Sep 04 '14 at 08:24
  • @DietrichBurde -- I guess I meant to let the OP be the judge of whether or not he was satisfied, but I apologize if my comment seemed impolite. I should clarify that the question I was saying would be good for MO is "why is it important that type A cyclotomic Hecke algebras admit a grading?". The general question about why it's useful for an algebra to be graded might be considered over-broad and probably wouldn't wind up addressing the OP's specific question. – Hugh Thomas Sep 04 '14 at 15:15

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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.

Dietrich Burde
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  • I agree that the paper you referenced is very relevant. Definition 2.2.1 does not define the cyclotomic quiver Hecke algebra; it's defined in 2.2.9. It is clear from definition 2.2.9 that it is graded, but it isn't clear that (as an ungraded algebra) it's isomorphic to the cyclotomic Hecke algebra. The fact that it is isomorphic is what means that Definition 2.2.9 amounts to giving a grading to the cyclotomic Hecke algebra. – Hugh Thomas Sep 04 '14 at 15:09
  • @HughThomas You are right, I confused the two definitions. It is definition $2.2.9$. – Dietrich Burde Sep 04 '14 at 15:17