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Tag: rationality

25 years monstrous moonshine

Writing a survey paper is a highly underestimated task. I once
tried it out with \’Centers of generic division algebras : the
rationality problem 1965-1990\’ and it took me a lot of time and that
was on a topic with only 10 to 15 key papers to consider… The task of
writing a survey paper on a topic with any breadth must be much more
difficult. Last week, Terry Gannon posted a survey paper on the arXiv :
Monstrous Moonshine : The first twenty-five years
which gives a very readable introduction to this exciting topic. It has
a marvelous opening line :

It has been approximately
twenty-five years since John McKay remarked that

196 884 = 196 883 +
 1

Anyone who is puzzled by this line (“So what?”)
should definitely have a go at this paper! Still not convinced? Here is
the second sentence :

That time has seen the discovery of
important structures, the establishment of another deep connection
between number theory and algebra, and a reinforcement of a new era of
cooperation between pure mathematics and mathematical
physics.

For the remaining sentences (quite a few, the paper
is 33 pages long) I happily refer you to the paper.

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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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