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!