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Tag: geometry

Brauer’s forgotten group

Non-commutative geometry seems pretty trivial compared
to commutative geometry : there are just two types of manifolds,
points and curves. However, nobody knows how to start classifying
these non-commutative curves. I do have a conjecture that any
non-commutative curve can (up to non-commutative birationality) be
constructed from hereditary orders over commutative curves
by universal methods but I’ll try to explain that another
time.

On the other hand, non-commutative points
have been classified (at least in principle) for at least 50
years over an arbitrary basefield $l$. non-commutative
$l$-points $P$ is an $l$-algebra such that its double
$d(P) = P \\otimes P^o$ ( where $P^o$ is the opposite algebra,
that is with the reverse multiplication) has an element$c=\\sum_i
a_i \\otimes b_i with \\sum_i a_ib_i = 1 (in $P$)$ and such that for
all p in $P$ we have that $(1 \\otimes a).c = (a \\otimes 1).c$ For
people of my generation, c is called a separability idempotent
and $P$ itself is called a separable $l$-algebra.
Examples of $l$-points include direct sums of full matrixrings
(of varying sizes) over $l$ or group-algebras $lG$ for $G$ a
finite group of n elements where n is invertible in $l$. Hence, in
particular, the group-algebra $lG$ of a p-group $G$ over a field $l$
of characteristic p is a non-commutative singular point and
modular representation theory (a theory build almost single
handed by
Richard Brauer) can be viewed as
the methods needed to resolve this singularity. Brauer’s name is
still mentioned a lot in modular representation theory, but another
of his inventions, the Brauer group, seems to be hardly known
among youngsters.

Still, it is a crucial tool
in classifying all non-commutative $l$-points. The algebraic
structure of an $l$-point $P$ is as follows : $$P = S_1 + S_2 + …
+ S_k$$ where each S_i is a simple algebra (that is, it
contains no proper twosided ideals), finite dimensional over
its center $l_i$ which is in its turn a finite dimensional
separable field extension of $l$. So we need to know all
simple algebras $S$, finite dimensional over their center $L$ which
is any finite dimensional separable field extension of $l$. The
algebraic structure of such an $S$ is of the form$$S = M(a,D)$$ that
is, full axa matrices with entries in $D$ where $D$ is a
skew-field (or some say, a division algebra) with
center $L$. The $L$-dimension of such a $D$ is always a square,
say b^2, so that the $L$-dimension of $S$ itself is also a square
a^2b^2. There are usually plenty such division algebras, the simplest
examples being quaternion algebras. Let p and q be two
non-zero elements of $L$ such that the conic $C : X^2-pY^2-bZ^2 =
0$ has no $L$-points in the projective $L$-plane, then the
algebra$D=(p,q)_2 = L.1 + L.i + L.j + L.ij where i^2=p, j^2=q and
ji=-ij$ is a four-dimensional skew-field over $L$. Brauer’s idea to
classify all simple $L$-algebras was to associate a group to them,
the Brauer group, $Br(L)$. Its elements are equivalence
classes
of simple algebras where two simple algebras $S$ and
$S’$ are equivalent if and only if$M(m,S) = M(n,S’)$ for some sizes
m and n. Multiplication on these classes in induced by
the tensor-product (over $L$) as $S_1 \\otimes S_2$ is again a simple
$L$-algebra if $S_1$ and $S_2$ are. The Brauer group $Br(L)$ is an
Abelian torsion group and if we know its structure we know all
$L$-simple algebras so if we know $Br(L)$ for all finite dimensional
separable extensions $L$ of $l$ we have a full classification of
all non-commutative $l$-points.

Here are some examples
of Brauer groups : if $L$ is algebraically closed (or separable
closed), then $Br(L)=0$ so in particular, if $l$ is algebraically
closed, then the only non-commutative points are sums of matrix rings.
If $R$ is the field of real numbers, then $Br(R) = Z/2Z$ generated by
the Hamilton quaternion algebra (-1,-1)_2. If $L$ is a complete
valued number field, then $Br(L)=Q/Z$ which allows to describe also
the Brauer group of a number field in terms of its places. Brauer groups
of function fields of (commutative) varieties over an algebraically
closed basefield is usually huge but there is one noteworthy
exception $Tsen’s theorem$ which states that $Br(L)=0$ if $L$ is the
function field of a curve C over an algebraically closed field. In 1982
Merkurjev and Suslin proved a marvelous result about generators of
$Br(L)$ whenever $L$ is large enough to contain all primitive roots
of unity. They showed, in present day lingo, that $Br(L)$
is generated by non-commutative points of the quantum-planes
over $L$ at roots of unity. That is, it is generated by cyclic
algebras
of the form$(p,q)_n = L
\\< X,Y>/(X^n=p,Y^n=q,YX=zXY)$where z is an n-th primitive root of
unity. Next time we will recall some basic results on the relation
between the Brauer group and Galois cohomology.

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NOG master class update


Yesterday I made a preliminary program for the first two months
of the masterclass non-commutative geometry. It is likely that
the program will still undergo changes as at the moment I included only
the mini-courses given by Bernhard
Keller
and Markus Reineke but several other people have
already agreed to come and give a talk. For example, Jacques Alev (Reims),
Tom Lenagan (Edinburgh),
Shahn Majid (London),
Giovanna Carnovale (Padua) among others. And in
may, Fred assures me, Maxim Kontsevich will give a couple of talks.

As for the contents of the two courses I will be
teaching I changed my mind slightly. The course non-commutative
geometry
I teach jointly with Markus Reineke and making the program
I realized that I have to teach the full 22 hours before he will start
his mini-course in the week of March 15-19 to explain the few
things
he needs, like :

To derive all the
counting of points formulas, I only need from your course:

the definition of formally smooth algebras basic properties, like
being
hereditary
– the definition of the component
semigroup
– the fact that dim Hom-dim Ext is constant along
components. This I need
even over finite fields $F_q$, but I
went through your proof in “One quiver”,
and it works. The
key fact is that even over $F_q$, the infinitesimal lifting
property implies smoothness in the sense Dimension of variety =
dimension of
(schematic) tangent space in any $F_q$-valued
point. But I think it’s fine for
the students if you do all
this over C, and I’ll only sketch the (few)
modifications for
algebras over $F_q$.

So my plan is to do all of
this first and leave the (to me) interesting problem of trying to
classify formally smooth algebras birationally to the second
course projects in non-commutative geometry which fits the title
as a lot of things still need to be done. The previous idea to give in
that course applications of non-commutative orders to the resolution of
singularities (in particular of quotient singularities) as very roughly
explained in my three talks on non-commutative geometry@n I now
propose to relegate to the friday afternoon seminar. I’ll be
happy to give more explanations on all this (in particular more
background on central simple algebras and the theory of (maximal)
orders) if other people work through the main part of the paper in the
seminar. In fact, all (other) suggestions for seminar-talks are welcome
: just tell me in person or post a comment to this post.

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Bill Schelter’s Maxima

Bill
Schelter was a remarkable man. First, he was a top-class mathematician.
If you allow yourself to be impressed, read his proof of the
Artin-Procesi theorem. Bill was also among the first to take
non-commutative geometry seriously. Together with Mike Artin he
investigated a notion of non-commutative integral extensions and he was
the first to focuss attention to formally smooth algebras (a
suggestion later taken up by a.o. Cuntz-Quillen and Kontsevich) and a
relative version with respect to algebras satisfying all identities of
n x n matrices which (via work of Procesi) led to smooth@n
algebras. To youngsters, he is probably best know as the co-inventor of
Artin-Schelter regular algebras. I still vividly remember an
overly enthusiastic talk by him on the subject in Oberwolfach, sometime
in the late eighties. Secondly, Bill was a genuine Lisp-guru and
a strong proponent of open source software, see for example his
petition against software patents. He maintanind
his own version of Kyoto Common Lisp which developed into Gnu
Common Lisp
. A quote on its history :

GCL is
the product of many hands over many years. The original effort was known
as the Kyoto Common Lisp system, written by Taiichi Yuasa and Masami
Hagiya in 1984. In 1987 new work was begun by William Schelter, and that
version of the system was called AKCL (Austin Kyoto Common Lisp). In
1994 AKCL was released as GCL (GNU Common Lisp) under the GNU public
library license. The primary purpose of GCL during that phase of it’s
existence was to support the Maxima computer algebra system, also
maintained by Dr. Schelter. It existed largely as a subproject of
Maxima.

Maxima started as Bill’s version of
Macsyma an MIT-based symbolic computation program to which he
added many routines, one of which was Affine a package that
allowed to do Groebner-like computations in non-commutative
algebras (implementing Bergman’s diamond lemma) and which he
needed to get a grip on 3-dimensional Artin-Schelter regular
algebras
. Michel and me convinced Fred to acquire funds to
buy us a work-station (costing at the time 20 to 30 iMacs) and have Bill
flown in from the States with his tape of maxima and let him
port it to our Dec-station. Antwerp was probably for years
the only place in the world (apart from MIT) where one could do
calculations in affine (probably highly illegal at the time).
Still, lots of people benefitted from this, among others Michaela
Vancliff
and Kristel Van Rompay in their investigation
of 4-dimensional Artin-Schelter regular algebras associated to an
automorphism of a quadric in three-dimensional projective space.
Yesterday I ran into Bill (alas virtually) by browsing the
crypto-category of Fink. There it was, maxima, Bill’s package! I tried to install it
with the Fink Commander and failed but succeeded from the command line.
So, if you want to have your own version of it type

sudo fink
install maxima

from the Terminal and it will install without
problems (giving you also a working copy of common lisp). Unfortunately
I do not remember too much of Macsyma or Affine but there is plenty of
documentation on the net. Manuals and user guides can be obtained from
the maxima homepage and the University of Texas
(Bill’s university) maintains an online manual, including a cryptic description of
some Affine-commands. But probably I’ll have to send Michaela an
email asking for some guidance on this… Here, as a tribute to Bill who
died in july 2001 the opening banner

 iMacLieven:~ lieven$
/sw/bin/maxima Maxima 5.9.0 http://maxima.sourceforge.net 
Distributed under the GNU Public License. 
See the file COPYING. 
Dedicated to the memory of William Schelter. 
This is a development version of Maxima. 
The function bug_report() provides bug reporting information. 
(C1)
 
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