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Graded Group, wait what?

  1. Feb 12, 2013 #1
    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!
  2. jcsd
  3. Feb 13, 2013 #2
    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:

    [tex]R=R_0\oplus R_1 \oplus R_2 \oplus ...[/tex]

    Furthermore, we demands that [itex]R_sR_t\subseteq R_{s+t}[/itex].

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

    Lets take [itex]R=k[X][/itex]. We can now define [itex]R_s=\{\alpha k^s~\vert~\alpha \in k\}[/itex]. So for example, [itex]2\in R_0[/itex], [itex]3X^3\in R_3[/itex] and [itex]3X+X^3[/itex] is not in any [itex]R_s[/itex]. 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 [itex]r\in R_s[/itex].

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

    A graded group is a very similar concept. But the original motivation comes from studying the polynomial ring and generalizing it to graded rings.
  4. Feb 13, 2013 #3
    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!
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