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driven by ambition and sloth

Published September 3, 2004 by lievenlb

Here’s a part of yesterday’s post by bitch ph.d. :

But first of all I have to figure out what the hell I’m going to teach my graduate students this semester, and really more to the point, what I am not going to bother to try to cram into this class just because it’s my first graduate class and I’m feeling like teaching everything I know in one semester is a realistic and desireable possibility. Yes! Here it all is! Everything I have ever learned! Thank you, and goodnight!

Ah, the perpetual motion machine of last-minute course planning, driven by ambition and sloth!.

I’ve had similar experiences, even with undergraduate courses (in Belgium there is no fixed curriculum so the person teaching the course is responsible for its contents). If you compare the stuff I hoped to teach when I started out with the courses I’ll be giving in a few weeks, you would be more than disappointed.
The first time I taught _differential geometry 1_ (a third year course) I did include in the syllabus everything needed to culminate in an outline of Donaldson’s result on exotic structures on $\mathbb{R}^4 $ and Connes’ non-commutative GUT-model (If you want to have a good laugh, here is the set of notes). As far as I remember I got as far as classifying compact surfaces!
A similar story for the _Lie theory_ course. Until last year this was sort of an introduction to geometric invariant theory : quotient variety of conjugacy classes of matrices, moduli space of linear dynamical systems, Hilbert schemes and the classification of $GL_n $-representations (again, smile! here is the set of notes).
Compared to these (over)ambitious courses, next year’s courses are lazy sunday-afternoon walks! What made me change my mind? I learned the hard way something already known to the ancient Greeks : mathematics does not allow short-cuts, you cannot expect students to run before they can walk. Giving an over-ambitious course doesn’t offer the students a quicker road to research, but it may result in a burn-out before they get even started!

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nog course outline

Published September 2, 2004 by lievenlb

Now that the preparation for my undergraduate courses in the first semester is more or less finished, I can begin to think about the courses I’ll give this year in the master class
non-commutative geometry. For a change I’d like to introduce the main ideas and concepts by a very concrete example : Ginzburg’s coadjoint-orbit result for the Calogero-Moser space and its
relation to the classification of one-sided ideals in the first Weyl algebra. Not only will this example give me the opportunity to say things about formally smooth algebras, non-commutative
differential forms and even non-commutative symplectic geometry, but it also involves what some people prefer to call _non-commutative algebraic geometry_ (that is the study of graded Noetherian
rings having excellent homological properties) via the projective space associated to the homogenized Weyl algebra. Besides, I have some affinity with this example.

A long time ago I introduced
the moduli spaces for one-sided ideals in the Weyl algebra in Moduli spaces for right ideals of the Weyl algebra and when I was printing a _very_ preliminary version of Ginzburg’s paper
Non-commutative Symplectic Geometry, Quiver varieties, and Operads (probably because he send a preview to Yuri Berest and I was in contact with him at the time about the moduli spaces) the
idea hit me at the printer that the right way to look at the propblem was to consider the quiver

$\xymatrix{\vtx{} \ar@/^/[rr]^a & & \vtx{} \ar@(u,ur)^x \ar@(d,dr)_y \ar@/^/[ll]^b} $

which eventually led to my paper together with Raf Bocklandt Necklace Lie algebras and noncommutative symplectic geometry.

Apart from this papers I would like to explain the following
papers by illustrating them on the above example : Michail Kapranov Noncommutative geometry based on commutator expansions Maxim Kontsevich and Alex Rosenberg Noncommutative smooth
spaces Yuri Berest and George Wilson Automorphisms and Ideals of the Weyl Algebra Yuri Berest and George Wilson Ideal Classes of the Weyl Algebra and Noncommutative Projective
Geometry Travis Schedler A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver and of course the seminal paper by Joachim Cuntz and Daniel Quillen on
quasi-free algebras and their non-commutative differential forms which, unfortunately, in not available online.

I plan to write a series of posts here on all this material but I will be very
happy to get side-tracked by any comments you might have. So please, if you are interested in any of this and want to have more information or explanation do not hesitate to post a comment (only
your name and email is required to do so, you do not have to register and you can even put some latex-code in your post but such a posting will first have to viewed by me to avoid cluttering of
nonsense GIFs in my directories).

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the Azumaya locus does determine the order

Published September 1, 2004 by lievenlb

Clearly
this cannot be correct for consider for $n \in \mathbb{N} $ the order

$A_n = \begin{bmatrix} \mathbb{C}[x] & \mathbb{C}[x] \\ (x^n) &
\mathbb{C}[x] \end{bmatrix} $

For $m \not= n $ the orders $A_n $
and $A_m $ have isomorphic Azumaya locus, but are not isomorphic as
orders. Still, the statement in the heading is _morally_ what Nikolaus
Vonessen and Zinovy
Reichstein are proving in their paper Polynomial identity
rings as rings of functions. So I better clarify what they do claim
precisely.

Let $A $ be a _Cayley-Hamilton order_, that is, a
prime affine $\mathbb{C} $-algebra, finite as a module over its center
and satisfying all trace relations holding in $M_n(\mathbb{C}) $. If $A $
is generated by $m $ elements, then its _representation variety_
$\mathbf{rep}_n~A $ has as points the m-tuples of $n \times n $ matrices

$(X_1,\ldots,X_m) \in M_n(\mathbb{C}) \oplus \ldots \oplus
M_n(\mathbb{C}) $

which satisfy all the defining relations of
A. $\mathbf{rep}_n~A $ is an affine variety with a $GL_n $-action
(induced by simultaneous conjugation in m-tuples of matrices) and has
as a Zariski open subset the tuples $(X_1,\ldots,X_m) \in
\mathbf{rep}_n~A $ having the property that they generate the whole
matrix-algebra $M_n(\mathbb{C}) $. This open subset is called the
Azumaya locus of A and denoted by $\mathbf{azu}_n~A $.

One can also define the _generic Azumaya locus_ as being the
Zariski open subset of $M_n(\mathbb{C}) \oplus \ldots \oplus
M_n(\mathbb{C}) $ consisting of those tuples which generate
$M_n(\mathbb{C}) $ and call this subset $\mathbf{Azu}_n $. In fact, one
can show that $\mathbf{Azu}_n $ is the Azumaya locus of a particular
order namely the trace ring of m generic $n \times n $ matrices.

What Nikolaus and Zinovy prove is that for an order A the Azumaya
locus $\mathbf{azu}_n~A $ is an irreducible subvariety of
$\mathbf{Azu}_n $ and that the embedding

$\mathbf{azu}_n~A
\subset \mathbf{Azu}_n $

determines A itself! If you have
worked a bit with orders this result is strange at first until you
recognize it as being essentially a consequence of Bill Schelter's
catenarity result for affine p.i.-algebras.

On the positive
side it shows that the study of orders is roughly equivalent to that of
the study of irreducible $GL_n $-stable subvarieties of $\mathbf{Azu}_n $.
On the negative side, it shows that the $GL_n $-structure of
$\mathbf{Azu}_n $ is horribly complicated. For example, it is still
unknown in general whether the quotient-variety (which is here also the
orbit space) $\mathbf{Azu}_n / GL_n $ is a rational variety.

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