Connection between polynomials and groups

In summary, the conversation discusses the connection between polynomials and groups in the context of exotic smoothness on manifolds. The paper mentions creating a line bundle over complex projective space and states that the elements of the line bundle are degree 2 polynomials. However, it is later clarified that the notation O(2) actually refers to a twisted sheaf. The conversation also mentions possible insights from Hatcher and encourages further contributions from others.
  • #1
cduston
8
0
Hey Everyone,
I'm reading a paper by Claude LeBrun about exotic smoothness on manifolds and he is talking about a connection between polynomials and groups that I am not familiar with (or at least I think that's what he's talking about). He's creating a line bundle (which happens to be O(2)) over the complex projective space CP^2 and he states that the "elements n of the line bundle are degree 2" (not actually a direct quote, but I'm almost positive that's what he's trying to say). So what this says to me is that there is some connection between the degree 2 polynomials (like p(x)=a+b*x + c*x^2) and the group of all 2x2 orthogonal matricies. I've been running this around in my head for a few days and I can't come up with a good explanation, so could someone help me out?

Thanks so much!
 
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  • #2
Are you sure O(2) is denoting an orthogonal group, and is not instead supposed to be the twisted sheaf [itex]\mathcal{O}(2)[/itex]? This latter interpretation sounds better to me, because if my understanding of algebraic geometry is correct, [itex]\mathcal{O}(2)[/itex] on the projective plane is supposed to be (roughly speaking) the set of all homogenous degree 2 polynomials in three variables.
 
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  • #3
Oh no...I think you are right, that's what the Os look like in the paper. I also had a conversation with my advisor about this but it was over e-mail, so I'm sure she wrote O(2) when she really meant (that crazy italic O)(2). Ok, it's back to Hatcher to learn about twisted sheafs! If anyone else has any insights they would be appreciated, but I think Hurkyl has the right idea. Thanks!
 
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  • #4
This is a quickie, and possibly wrong since it is what I think of but I'm no geometer.

The projective plane is just two copies of the affine plane glued together. You can think of the coords on each plane as, say,

k[x,y] and k[u,v]

We glue them, roughly speaking by sending x to u, and y to v^-n, you think of this as twisting n times (n can be positive or negative here), and this is O(n). (The font is mathcal, I think).
 

1. What is the connection between polynomials and groups?

The connection between polynomials and groups is that polynomials can be used to construct groups. Specifically, the coefficients of a polynomial can be used as the elements of a group, and the operations of addition and multiplication of polynomials can be used to define the group's binary operations.

2. How can polynomials be used to study groups?

Polynomials can be used to study groups by providing a visual representation of the group's structure. The roots of a polynomial correspond to the elements of the group, and the degree of the polynomial can provide information about the group's order and other properties.

3. Can polynomials be used to classify groups?

Yes, polynomials can be used to classify groups. The type of polynomial used to construct a group can determine its group structure, and different types of polynomials can lead to different types of groups, such as cyclic, abelian, or non-abelian groups.

4. How do polynomials relate to group theory?

Polynomials are an important tool in group theory as they can be used to define and analyze group structures. In particular, they are useful in studying finite groups and their properties, such as subgroups, cosets, and isomorphisms.

5. Are there any real-world applications of the connection between polynomials and groups?

Yes, there are several real-world applications of the connection between polynomials and groups. One example is in cryptography, where polynomials and their corresponding groups are used in encryption algorithms. Another example is in coding theory, where polynomials are used to construct error-correcting codes.

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