Specific (algebraic) directions in NCG

Question

Since my initial question (How much algebra is there in Noncommutative Geometry?), I got to study the basics of NCG and I now consider starting a PhD. program in NCG. I read Khalkhali's Basic NCG and to some extent, Landi's An Introduction to noncommutative spaces and their geometries. So I think it's fair to say that I got a bit of a general idea on the concerns of this field.

Now, since Khalkhali's book is more of a survey and most introductory books are like this, how can I continue with something deeper? What I mean is that I think I can easily get lost if I try to read $simultaneously$ some operator algebra, K-theory, differential geometry or any other subject that's included in NCG. I'd like to focus on a particular section, this being the best way to progress, but I can't seem to be able to direct myself.

Of course I'll have this talk with my future PhD. advisor, but your opinions are more than welcome.

I add that my approach is more algebraic, rather than Connes' analytic one. I also read Quillen and Cuntz article - Algebra Extensions and Nonsingularity, some Hochschild and cyclic homology articles and the first half of Kassel's Quantum Groups.

So...how can I bring order to this chaos, by not spreading between so many areas involved? What are exactly the possible algebraic directions? For the analytic/functional ones, it's clear that one has to read a C*-algebra book, for K-theory there are Karoubi, Bass, Rosenberg and others. But what about the algebraic part? What is it, exactly and where to find it?

Thank you very much for your time.

Answer 1

Now that I have a better sense of what you are looking for, I'll mention a few other references that may be of interest to you. I will certainly mention more here than you can reasonably be expected to read in the near future, since you're just starting your studies. But my feeling is that it's always helpful to be aware of more resources than you are currently using!

You already mentioned that you've read Cuntz-Quillen's paper and started learning about Calabi-Yau algebras; I want to verify that these sound like great topics to explore given your interests. You might also wish to take a look at the following textbook, which will have much more to say about differential algebra:

• J.-L. Loday, Cyclic Homology

There are a number of online lecture notes related to the "algebraic versions" of noncommutative geometry. Here are two that you might find interesting since you care about differential algebra and CY algebras:

• V. Ginzburg, Lectures on Noncommutative Geometry
• T. Schedler, Deformations of algebras in noncommutative geometry

The second set of notes came out of a summer school that was was associated with a recent special semester at MSRI, titled Noncommutative Algebraic Geometry and Representation Theory. Follow both of those links to find videos of various lectures, as well as a bibliography! (Incidentally, there is another set of notes by M. Wemyss, Lectures on Noncommutative Resolutions, that also came out of that summer school.)

If you are interested in the "algebraic part" of noncommutative geometry, this will likely lead you into noncommutative algebraic geometry. Web searches will return any number of lecture notes, from which you will quickly learn that there are many different approaches to this "subject." One of the more well-known branches is noncommutative projective geometry, which studies graded noncommutative algebras. An nice overview of the field can be found in Stafford and Van den Bergh's article Noncommutative curves and noncommutative surfaces. Though this is unfortunately now over a decade old, it's still a nice place to start reading about this topic.

Finally, you might find it useful to have a number of other pointers to the literature on Calabi-Yau algebras. Without adding any commentary, I have found the following articles particularly useful. You can find many more interesting articles by combing through their collective bibliographies.

• R. Berger and R. Taillefer, Poincare-Birkhoff-Witt deformations of Calabi-Yau algebras
• R. Bocklandt, Graded Calabi Yau algebras of dimension 3
• R. Bocklandt, T. Schedler, and M. Wemyss, Superpotentials and higher order derivations
• B. Keller, Calabi-Yau triangulated categories
• U. Kraehmer, Poincare duality in Hochschild (co)homology

Answer 2

As I said in the comment, I'll post an answer, since in the meantime I got to talk to a researcher in the field and somehow made my thoughts clearer. If there aren't any other answers, complaints or relevant comments, I'll accept this in a couple of days and call it a close.

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My initial thought was that C*-algebras, vector bundles and the like, which are contained in the "operator algebra" or the "NC topology" parts are somehow distinct from what I called the "algebraic part" (a vague term which I'll explain better below). I couldn't see connections between them and although NGC is a "unit", I thought these approaches were not interacting that much. Since I am somehow formed in (NC) algebra and I know little about functional analysis, algebraic or differential geometry, the "algebraic part" appealed more and seemed easier for me to grasp. That's why I needed more directions in this section.

I started reading a great book, "Elements of Noncommutative Geometry", by J. Varilly, H. Figueroa and J.M. Gracia-Bondia, which basically details all the approaches on NCG, starting with NC spaces, showing the basics of NC topology, operator algebras and their interactions and continues with K-theory, NC calculus and so forth. This book helped me a lot in seeing the somehow different approaches on the subject and helped me clarify and distinguish between the directions involved.

Now, what I mean by "algebraic part", based on my knowledge and owing greatly to the book above, is comprised of:

• algebraic K-theory, but not in its full generality, but rather its connections with NCG;

• NC differential algebras and forms - in this respect, I read Quillen and Cuntz's article - Algebra Extensions and Nonsingularity, which I found interesting, but I don't know where it's gone. I mean I know no further developments on NC differential forms;

• (compact) quantum groups and Hopf $*$-algebras. For this, I found the survey of Tuset and Kustermans, about the C*-algebraic quantum groups, Woronowicz's article on Compact Quantum Groups, the article of Vaes and van Daele on Hopf C*-algebras and other chapters in books on Quantum groups related to this.

Moreover, I found some interesting and new (for me) types of algebras, useful for NCG, namely the Calabi-Yau algebras and for this I found the article of Ginzburg, titled "Calabi-Yau algebras". I found that they were quite developed and used in various directions and I intend to study at least their basic properties.

So, for the moment, I am satisfied with the (compact) quantum groups and Hopf $*$-algebras that I found and I intend on studying them thoroughly. They appeal the most to me, in this state of knowledge that I find myself.

P.S. I know that Hopf $*$-algebras are strongly related to functional analysis, which I'm not fond of, but I think that this approach, via quantum groups and Hopf algebras makes things clearer and somehow easier for me to understand.