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Tag: Grothendieck

a cosmic Galois group

there hidden relations between mathematical and physical constants such

$\frac{e^2}{4 \pi \epsilon_0 h c} \sim \frac{1}{137} $

or are these numerical relations mere accidents? A couple of years
ago, Pierre Cartier proposed in his paper A mad day’s work : from Grothendieck to Connes and
Kontsevich : the evolution of concepts of space and symmetry
there are many reasons to believe in a cosmic Galois group acting on the
fundamental constants of physical theories and responsible for relations
such as the one above.

The Euler-Zagier numbers are infinite
sums over $n_1 > n_2 > ! > n_r \geq 1 $ of the form

$\zeta(k_1,\dots,k_r) = \sum n_1^{-k_1} \dots n_r^{-k_r} $

and there are polynomial relations with rational coefficients between
these such as the product relation

$\zeta(a)\zeta(b)=\zeta(a+b)+\zeta(a,b)+\zeta(b,a) $

It is
conjectured that all polynomial relations among Euler-Zagier numbers are
consequences of these product relations and similar explicitly known
formulas. A consequence of this conjecture would be that
$\zeta(3),\zeta(5),\dots $ are all trancendental!

introduced the Grothendieck-Teichmuller group-scheme over $\mathbb{Q} $
whose Lie algebra $\mathfrak{grt}_1 $ is conjectured to be the free Lie
algebra on infinitely many generators which correspond in a natural way
to the numbers $\zeta(3),\zeta(5),\dots $. The Grothendieck-Teichmuller
group itself plays the role of the Galois group for the Euler-Zagier
numbers as it is conjectured to act by automorphisms on the graded
$\mathbb{Q} $-algebra whose degree $d $-term are the linear combinations
of the numbers $\zeta(k_1,\dots,k_r) $ with rational coefficients and
such that $k_1+\dots+k_r=d $.

The Grothendieck-Teichmuller
group also appears mysteriously in non-commutative geometry. For
example, the set of all Kontsevich deformation quantizations has a
symmetry group which Kontsevich conjectures to be isomorphic to the
Grothendieck-Teichmuller group. See section 4 of his paper Operads and motives in
deformation quantzation
for more details.

It also appears
in the renormalization results of Alain Connes and Dirk Kreimer. A very
readable introduction to this is given by Alain Connes himself in Symmetries Galoisiennes
et renormalisation
. Perhaps the latest news on Cartier’s dream of a
cosmic Galois group is the paper by Alain Connes and Matilde Marcolli posted
last month on the arXiv : Renormalization and
motivic Galois theory
. A good web-page on all of this, including
references, can be found here.

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Brauer-Severi varieties

Classical Brauer-Severi varieties can be described either as twisted
forms of projective space (Severi\’s way) or as varieties containing
splitting information about central simple algebras (Brauer\’s way). If
$K$ is a field with separable closure $\overline{K}$, the first approach
asks for projective varieties $X$ defined over $K$ such that over the
separable closure $X(\overline{K}) \simeq
\mathbb{P}^{n-1}_{\overline{K}}$ they are just projective space. In
the second approach let $\Sigma$ be a central simple $K$-algebra and
define a variety $X_{\Sigma}$ whose points over a field extension $L$
are precisely the left ideals of $\Sigma \otimes_K L$ of dimension $n$.
This variety is defined over $K$ and is a closed subvariety of the
Grassmannian $Gr(n,n^2)$. In the special case that $\Sigma = M_n(K)$ one
can use the matrix-idempotents to show that the left ideals of dimension
$n$ correspond to the points of $\mathbb{P}^{n-1}_K$. As for any central
simple $K$-algebra $\Sigma$ we have that $\Sigma \otimes_K \overline{K}
\simeq M_n(\overline{K})$ it follows that the varieties $X_{\Sigma}$ are
among those of the first approach. In fact, there is a natural bijection
between those of the first approach (twisted forms) and of the second as
both are classified by the Galois cohomology pointed set
$H^1(Gal(\overline{K}/K),PGL_n(\overline{K}))$ because
$PGL_n(\overline{K})$ is the automorphism group of
$\mathbb{P}^{n-1}_{\overline{K}}$ as well as of $M_n(\overline{K})$. The
ringtheoretic relevance of the Brauer-Severi variety $X_{\Sigma}$ is
that for any field extension $L$ it has $L$-rational points if and only
if $L$ is a _splitting field_ for $\Sigma$, that is, $\Sigma \otimes_K L
\simeq M_n(\Sigma)$. To give one concrete example, If $\Sigma$ is the
quaternion-algebra $(a,b)_K$, then the Brauer-Severi variety is a conic
$X_{\Sigma} = \mathbb{V}(x_0^2-ax_1^2-bx_2^2) \subset \mathbb{P}^2_K$
Whenever one has something working for central simple algebras, one can
_sheafify_ the construction to Azumaya algebras. For if $A$ is an
Azumaya algebra with center $R$ then for every maximal ideal
$\mathfrak{m}$ of $R$, the quotient $A/\mathfrak{m}A$ is a central
simple $R/\mathfrak{m}$-algebra. This was noted by the
sheafification-guru [Alexander Grothendieck][2] and he extended the
notion to Brauer-Severi schemes of Azumaya algebras which are projective
bundles $X_A \rightarrow \mathbf{max}~R$ all of which fibers are
projective spaces (in case $R$ is an affine algebra over an
algebraically closed field). But the real fun started when [Mike
Artin][3] and [David Mumford][4] extended the construction to suitably
_ramified_ algebras. In good cases one has that the Brauer-Severi
fibration is flat with fibers over ramified points certain degenerations
of projective space. For example in the case considered by Artin and
Mumford of suitably ramified orders in quaternion algebras, the smooth
conics over Azumaya points degenerate to a pair of lines over ramified
points. A major application of their construction were examples of
unirational non-rational varieties. To date still one of the nicest
applications of non-commutative algebra to more mainstream mathematics.
The final step in generalizing Brauer-Severi fibrations to arbitrary
orders was achieved by [Michel Van den Bergh][5] in 1986. Let $R$ be an
affine algebra over an algebraically closed field (say of characteristic
zero) $k$ and let $A$ be an $R$-order is a central simple algebra
$\Sigma$ of dimension $n^2$. Let $\mathbf{trep}_n~A$ be teh affine variety
of _trace preserving_ $n$-dimensional representations, then there is a
natural action of $GL_n$ on this variety by basechange (conjugation).
Moreover, $GL_n$ acts by left multiplication on column vectors $k^n$.
One then considers the open subset in $\mathbf{trep}_n~A \times k^n$
consisting of _Brauer-Stable representations_, that is those pairs
$(\phi,v)$ such that $\phi(A).v = k^n$ on which $GL_n$ acts freely. The
corresponding orbit space is then called the Brauer-Severio scheme $X_A$
of $A$ and there is a fibration $X_A \rightarrow \mathbf{max}~R$ again
having as fibers projective spaces over Azumaya points but this time the
fibration is allowed to be far from flat in general. Two months ago I
outlined in Warwick an idea to apply this Brauer-Severi scheme to get a
hold on desingularizations of quiver quotient singularities. More on
this next time.


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quiver representations

In what
way is a formally smooth algebra a _machine_ producing families of
manifolds? Consider the special case of the path algebra $\mathbb{C} Q$ of a
quiver and recall that an $n$-dimensional representation is an algebra
map $\mathbb{C} Q \rightarrow^{\phi} M_n(\mathbb{C})$ or, equivalently, an
$n$-dimensional left $\mathbb{C} Q$-module $\mathbb{C}^n_{\phi}$ with the action
determined by the rule $a.v = \phi(a) v~\forall v \in \mathbb{C}^n_{\phi},
\forall a \in \mathbb{C} Q$ If the $e_i~1 \leq i \leq k$ are the idempotents
in $\mathbb{C} Q$ corresponding to the vertices (see this [post][1]) then we get
a direct sum decomposition $\mathbb{C}^n_{\phi} = \phi(e_1)\mathbb{C}^n_{\phi} \oplus
\ldots \oplus \phi(e_k)\mathbb{C}^n_{\phi}$ and so every $n$-dimensional
representation does determine a _dimension vector_ $\alpha =
(a_1,\ldots,a_k)~\text{with}~a_i = dim_{\mathbb{C}} V_i = dim_{\mathbb{C}}
\phi(e_i)\mathbb{C}^n_{\phi}$ with $ | \alpha | = \sum_i a_i = n$. Further,
for every arrow $\xymatrix{\vtx{e_i} \ar[rr]^a & &
\vtx{e_j}} $ we have (because $e_j.a.e_i = a$ that $\phi(a)$
defines a linear map $\phi(a)~:~V_i \rightarrow V_j$ (that was the
whole point of writing paths in the quiver from right to left so that a
representation is determined by its _vertex spaces_ $V_i$ and as many
linear maps between them as there are arrows). Fixing vectorspace bases
in the vertex-spaces one observes that the space of all
$\alpha$-dimensional representations of the quiver is just an affine
space $\mathbf{rep}_{\alpha}~Q = \oplus_a~M_{a_j \times a_i}(\mathbb{C})$ and
base-change in the vertex-spaces does determine the action of the
_base-change group_ $GL(\alpha) = GL_{a_1} \times \ldots \times
GL_{a_k}$ on this space. Finally, as all this started out with fixing
a bases in $\mathbb{C}^n_{\phi}$ we get the affine variety of all
$n$-dimensional representations by bringing in the base-change
$GL_n$-action (by conjugation) and have $\mathbf{rep}_n~\mathbb{C} Q =
\bigsqcup_{| \alpha | = n} GL_n \times^{GL(\alpha)}
\mathbf{rep}_{\alpha}~Q$ and in this decomposition the connected
components are no longer just affine spaces with a groupaction but
essentially equal to them as there is a natural one-to-one
correspondence between $GL_n$-orbits in the fiber-bundle $GL_n
\times^{GL(\alpha)} \mathbf{rep}_{\alpha}~Q$ and $GL(\alpha)$-orbits in the
affine space $\mathbf{rep}_{\alpha}~Q$. In our main example
$\xymatrix{\vtx{e} \ar@/^/[rr]^a & & \vtx{f} \ar@(u,ur)^x
\ar@(d,dr)_y \ar@/^/[ll]^b} $ an $n$-dimensional representation
determines vertex-spaces $V = \phi(e) \mathbb{C}^n_{\phi}$ and $W = \phi(f)
\mathbb{C}^n_{\phi}$ of dimensions $p,q~\text{with}~p+q = n$. The arrows
determine linear maps between these spaces $\xymatrix{V
\ar@/^/[rr]^{\phi(a)} & & W \ar@(u,ur)^{\phi(x)} \ar@(d,dr)_{\phi(y)}
\ar@/^/[ll]^{\phi(b)}} $ and if we fix a set of bases in these two
vertex-spaces, we can represent these maps by matrices
$\xymatrix{\mathbb{C}^p \ar@/^/[rr]^{A} & & \mathbb{C}^q \ar@(u,ur)^{X}
\ar@(d,dr)_{Y} \ar@/^/[ll]^{B}} $ which can be considered as block
$n \times n$ matrices $a \mapsto \begin{bmatrix} 0 & 0 \\ A & 0
\end{bmatrix}~b \mapsto \begin{bmatrix} 0 & B \\ 0 & 0 \end{bmatrix}$
$x \mapsto \begin{bmatrix} 0 & 0 \\ 0 & X \end{bmatrix}~y \mapsto
\begin{bmatrix} 0 & 0 \\ 0 & Y \end{bmatrix}$ The basechange group
$GL(\alpha) = GL_p \times GL_q$ is the diagonal subgroup of $GL_n$
$GL(\alpha) = \begin{bmatrix} GL_p & 0 \\ 0 & GL_q \end{bmatrix}$ and
acts on the representation space $\mathbf{rep}_{\alpha}~Q = M_{q \times
p}(\mathbb{C}) \oplus M_{p \times q}(\mathbb{C}) \oplus M_q(\mathbb{C}) \oplus M_q(\mathbb{C})$
(embedded as block-matrices in $M_n(\mathbb{C})^{\oplus 4}$ as above) by
simultaneous conjugation. More generally, if $A$ is a formally smooth
algebra, then all its representation varieties $\mathbf{rep}_n~A$ are
affine smooth varieties equipped with a $GL_n$-action. This follows more
or less immediately from the definition and [Grothendieck][2]\’s
characterization of commutative regular algebras. For the record, an
algebra $A$ is said to be _formally smooth_ if for every algebra map $A
\rightarrow B/I$ with $I$ a nilpotent ideal of $B$ there exists a lift
$A \rightarrow B$. The path algebra of a quiver is formally smooth
because for every map $\phi~:~\mathbb{C} Q \rightarrow B/I$ the images of the
vertex-idempotents form an orthogonal set of idempotents which is known
to lift modulo nilpotent ideals and call this lift $\psi$. But then also
every arrow lifts as we can send it to an arbitrary element of
$\psi(e_j)\pi^{-1}(\phi(a))\psi(e_i)$. In case $A$ is commutative and
$B$ is allowed to run over all commutative algebras, then by
Grothendieck\’s criterium $A$ is a commutative regular algebra. This
also clarifies why so few commutative regular algebras are formally
smooth : being formally smooth is a vastly more restrictive property as
the lifting property extends to all algebras $B$ and whenever the
dimension of the commutative variety is at least two one can think of
maps from its coordinate ring to the commutative quotient of a
non-commutative ring by a nilpotent ideal which do not lift (for an
example, see for example [this preprint][3]). The aim of
non-commutative algebraic geometry is to study _families_ of manifolds
$\mathbf{rep}_n~A$ associated to the formally-smooth algebra $A$. [1]: [2]:
html [3]:

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noncommutative geometry 2

Again I
spend the whole morning preparing my talks for tomorrow in the master
class. Here is an outline of what I will cover :
– examples of
noncommutative points and curves. Grothendieck’s characterization of
commutative regular algebras by the lifting property and a proof that
this lifting property in the category alg of all l-algebras is
equivalent to being a noncommutative curve (using the construction of a
generic square-zero extension).
– definition of the affine
scheme rep(n,A) of all n-dimensional representations (as always,
l is still arbitrary) and a proof that these schemes are smooth
using the universal property of k(rep(n,A)) (via generic
– whereas rep(n,A) is smooth it is in general
a disjoint union of its irreducible components and one can use the
sum-map to define a semigroup structure on these components when
l is algebraically closed. I’ll give some examples of this
semigroup and outline how the construction can be extended over
arbitrary basefields (via a cocommutative coalgebra).

definition of the Euler-form on rep A, all finite dimensional
representations. Outline of the main steps involved in showing that the
Euler-form defines a bilinear form on the connected component semigroup
when l is algebraically closed (using Jordan-Holder sequences and
upper-semicontinuity results).

After tomorrow’s
lectures I hope you are prepared for the mini-course by Markus Reineke on non-commutative Hilbert schemes
next week.

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a noncommutative Grothendieck topology

We have seen that a non-commutative $l$-point is an
algebra$P=S_1 \\oplus … \\oplus S_k$with each $S_i$ a simple
finite dimensional $l$-algebra with center $L_i$ which is a separable
extension of $l$. The centers of these non-commutative points (that is
the algebras $L_1 \\oplus … \\oplus L_k$) are the open sets of a
Grothendieck-topology on
$l$. To define it properly, let $L$ be the separable closure of $l$
and let $G=Gal(L/l)$ be the so called absolute
Galois group. Consider the
category with objects the finite $G$-sets, that is : finite
sets with an action of $G$, and with morphisms the $G$-equivariant
set-maps, that is: maps respecting the group action. For each object
$V$ we call a finite collection of morphisms $Vi \\mapsto V$ a
cover of $V$ if the images of the finite number of $Vi$ is all
of $V$. Let $Cov$ be the set of all covers of finite $G$-sets, then
this is an example of a Grothendieck-topology as it satisfies
the following three conditions :

(GT1) : If
$W \\mapsto V$ is an isomorphism of $G$-sets, then $\\{ W \\mapsto
V \\}$ is an element of $Cov$.

(GT2) : If $\\{ Vi \\mapsto
V \\}$ is in $Cov$ and if for every i also $\\{ Wij \\mapsto Vi \\}$
is in $Cov$, then the collection $\\{ Wij \\mapsto V \\}$ is in

(GT3) : If $\\{ fi : Vi \\mapsto V \\}$ is in $Cov$
and $g : W \\mapsto V$ is a $G$-morphism, then the fibered
$Vi x_V W = \\{ (vi,w) in Vi x W : fi(vi)=g(w) \\}$is
again a $G$-set and the collection $\\{ Vi x_V W \\mapsto V \\}$
is in $Cov$.

Now, finite $G$-sets are just
commutative separable $l$-algebras (that is,
commutative $l$-points). To see this, decompose a
finite $G$-set into its finitely many orbits $Oj$ and let $Hj$ be the
stabilizer subgroup of an element in $Oj$, then $Hj$ is of finite
index in $G$ and the fixed field $L^Hj$ is a finite dimensional
separable field extension of $l$. So, a finite $G$-set $V$
corresponds uniquely to a separable $l$-algebra $S(V)$. Moreover, a
finite cover $\\{ W \\mapsto V \\}$ is the same thing as saying
that $S(W)$ is a commutative separable $S(V)$-algebra. Thus,
the Grothendieck topology of finite $G$-sets and their covers
is anti-equivalent to the category of commutative separable
$l$-algebras and their separable commutative extensions.

This raises the natural question : what happens if we extend the
category to all separable $l$-algebras, that is, the category of
non-commutative $l$-points, do we still obtain something like a
Grothendieck topology? Or do we get something like a
non-commutative Grothendieck topology as defined by Fred Van
Oystaeyen (essentially replacing the axiom (GT 3) by a left and right
version). And if so, what are the non-commutative covers?
Clearly, if $S(V)$ is a commutative separable $l$-algebras, we expect
these non-commutative covers to be the set of all separable
$S(V)$-algebras, but what are they if $S$ is itself non-commutative,
that is, if $S$ is a non-commutative $l$-point?

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