What Are Graded Groups and How Do Generators and Degrees Function Within Them?

[Tenshou]

What exactly is a graded group, Is it just the direct decomp. of the group, or space? is it a way of breaking a group/space into its generators? how do these entities work? Help, please!

 

[micromass]

I think the easiest thing is if you know first what a graded ring is. It is easier because it has a very natural example. A graded ring is just a ring R which we can decompose in groups as follows:

R=R_0\oplus R_1 \oplus R_2 \oplus ...

Furthermore, we demands that R_sR_t\subseteq R_{s+t}.

The idea of a graded ring is to generalize one very important example, namely the polynomial ring.

Lets take R=k[X]. We can now define R_s=\{\alpha k^s~\vert~\alpha \in k\}. So for example, 2\in R_0, 3X^3\in R_3 and 3X+X^3 is not in any R_s. You can easily check the axioms for a graded ring now. The idea behind a graded ring is to define a certain "degree". Indeed, we say that r has degree s if r\in R_s.

Quite similarly, we can do the same for the multivariate polynomial rings. For example k[X,Y]. We define the degree of X^sY^t as s+t. Then we can again split up the ring k[X,Y]. For example XY\in R_2, X^4\in R_4, XY+X^2\in R_2 and XY+X^4 in no R_s.

A graded group is a very similar concept. But the original motivation comes from studying the polynomial ring and generalizing it to graded rings.

 

[Tenshou]

OH My... Thank you so much! you just open a huge place of exploration to me, I didn't quite get what they were saying, in the books when they talked about the degree. I wasn't sure if it was talking about the field and how far to which a degree they extended it(but I guess it can be used in that way also, now), or the degree of a polynomial, thank you so much! I will find it much more easier to get through this sections in this book!