Is pgl(2,Z) considered a "quantum group"?

Forum for the GRE subject test in mathematics.
yoyostein
Posts: 36
Joined: Tue Feb 28, 2012 12:14 am

Is pgl(2,Z) considered a "quantum group"?

Postby yoyostein » Tue Mar 13, 2012 12:29 am

Hi,

May I ask if the group pgl(2,Z) is a quantum group?

If yes/no, could briefly state the reason?

Sincerest thanks!

(Reason I am asking this is that my supervisor suggested the topic of quantum groups for my undergrad thesis. I don't have to prove any new theorems or new results but do have to show some "originality". Any ideas would be welcome.)

DanielMcLaury
Posts: 12
Joined: Mon Nov 21, 2011 10:42 pm

Re: Is pgl(2,Z) considered a "quantum group"?

Postby DanielMcLaury » Tue Mar 13, 2012 1:52 am

"Quantum groups" aren't actually groups; they're Hopf algebras.

*** Warning: not an expert by any means ***

In a lot of contexts in math it's useful to be able to "deform" objects. What this means, in some sense, is that you can consider the family of such objects as some kind of continuous geometric object, and you can move from one object to another via continuous paths. Since you bring up PSL(2, Z), I'm guessing you know about j-invariants, for instance.

The problem is that certain types of objects, e.g. compact groups, are "rigid" -- the geometric space they form is discrete, so you can't imagine a "path" through them that lets you slightly deform them. However, if you take a group algebra (which has a natural Hopf algebra structure), it can be deformed as a Hopf algebra.

Of course, almost none of the Hopf algebras in a neighborhood of a group algebra will be group algebras themselves, but the idea is to image that they are "group algebras of quantum groups." (In reality, there is no such object as a quantum group "downstairs").

yoyostein
Posts: 36
Joined: Tue Feb 28, 2012 12:14 am

Re: Is pgl(2,Z) considered a "quantum group"?

Postby yoyostein » Tue Mar 13, 2012 10:13 am

Thanks for the help! I would have to learn more about Hopf algebras then..

I read the description on Princeton companion to Mathematics:
"for example, that the alternating group A4 is naturally Ricci-flat, while the symmetric group S3 naturally has constant curvature [III.13], much like a 3-sphere."

(http://books.google.com.sg/books?id=ZOf ... re&f=false)

I wonder if the curvature would be reasonably possible to compute. If so, I could compute the curvature of some group as an application of Quantum Groups, which could possibly count as "original" application.

Any ideas on other possible applications of quantum groups?




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