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**Morita Equivalence Revisited**

**Holomorphic Structures on the Quantum Projective Line**

The three authors of this work, Masoud Khalkhali, Gianni Landi and Walter Van Suijlekom, have started very recently this joint work and are in an “exploratory” stage with this q-deformation of the two sphere. It is a very interesting concrete case to discover how much of the miraculous structure of the two sphere as a complex curve, or equivalently as a conformal manifold, actually survive the q-deformation. Many tools of NCG are available there, including the abstract perturbation of conformal structures by Beltrami differentials as explained in Chapter 4, section 4, pages 339-346 of the ncg book. A great challenge is to prove an analogue of the measurable Riemann mapping theorem in the q-deformed case. The formalism of q-groups allows one to set-up a simple algebraic framework but the real challenge resides in the analysis.

That was all for the talks of the morning session. The first talk in the afternoon was given by David Goss with abstract and title

**The group S(q) and indications of functional equations in finite characteristic**

**The quantum dilogarithm and quantization of cluster varieties **The subject of the talk is the joint work of Sacha Goncharov with V. V. Fock. The talk was excellent but it created a quite uneasy feeling in me which I had a hard time to identify. At first I thought it was due to the usual difficulty I have to hear a talk on something called the “quantum torus” and which looked like a reincarnation of the work I had done in 1980 on the representation of the noncommutative torus in L^2(R) and on the duality I had discovered there between the torus for theta and 1/theta (cf line 10 of page 8 of the english translation of the note). But in fact this “reincarnation” was appearing in a very strange way, with a factor of i=square root(-1) in the exponents of the operators acting on L^2(R) and that was the real reason why I felt disturbed. What I have found since then, and checked in an email exchange with Sacha, is that the commutation relations of these self-adjoint operators are only “formal” and hold on a dense domain but these operators actually **do not commute**. If you take the simplest case where q=1 then the presentation of the “quantum torus” is simply:

**A=A*, B=B* and AB=BA**———————————

**Essential dimension**

Even though my understanding of noncommutative geometry is limited, there are some aspects that I am able to follow.

I was wondering, since there are so few blogs here, why don't you guys forge an alliance with neverending books, you blog about noncommutative geometry anyways. That way you have another(n-category cafe) blogspot and gives well informed views(well depending on how well defined a conversational-style blog can be).

just for the record :

i did propose the very same idea to Alain Connes a while ago, and, though his initial response was pretty positive, he left the details to his minions.

i'm happy to share the full email correspondence, but that might be counter-productive. short story : nothing came of it.

i'm still very much in favour of your suggestion. but then, it takes two to tango…

Instead of whining about it, i've set up an open site where anyone who wants to share something about noncommutative geometry gets automatic author privileges and can post and comment unmoderated.

Here's the URL the n-geometry cafe.

Please read the FAQ to get you going. Enjoy and contribute!

@lieven le bruyn

Thanks for the link. I gonna study it.

Excuse my asking, but how is the operator B defined when f is in L^2 of R? What is f(x + 2*pi*i) if f is defined on R? What is the dense domain being referred to?