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Tag: Quillen

Segalโ€™s formal neighbourhood result

Yesterday, Ed Segal gave a talk at the Arts. His title “Superpotential algebras from 3-fold singularities” didnt look too promising to me. And sure enough it was all there again : stringtheory, D-branes, Calabi-Yaus, superpotentials, all the pseudo-physics babble that spreads virally among the youngest generation of algebraists and geometers.

Fortunately, his talk did contain a general ringtheoretic gem. After a bit of polishing up this gem, contained in his paper The A-infinity Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus, can be stated as follows.

Let $A $ be a $\mathbb{C} $-algebra and let $M = S_1 \oplus \ldots \oplus S_k $ be a finite dimensional semi-simple representation with distinct simple components. Let $\mathfrak{m} $ be the kernel of the algebra epimorphism $A \rightarrow S $ to the semi-simple algebra $S=End(M) $. Then, the $\mathfrak{m} $-adic completion of $A $ is Morita-equivalent to the completion of a quiver-algebra with relations. The nice thing is that both the quiver and relations come in a canonical way from the $A_{\infty} $-structure on the Ext-algebra $Ext^{\bullet}_A(M,M) $. More precisely, there is an isomorphism

$\hat{A}_{\mathfrak{m}} \simeq \frac{\hat{T}_S(Ext^1_A(M,M)^{\ast})}{(Im(HMC)^{\ast})} $

where the homotopy Maurer-Cartan map comes from the $A_{\infty} $ structure maps

$HMC = \oplus_i m_i~:~T_S(Ext_A^1(M,M)) \rightarrow Ext^2_A(M,M) $

and hence the defining relations of the completion are given by the image of the dual of this map.

For ages, Ive known this result in the trivial case of formally smooth algebras (where $Ext^2_A(M,M)=0 $ and hence there are no relations to divide out) and where it is a consequence of a special case of the Cuntz-Quillen “tubular neighborhood” result. Completions of formally smooth algebras at semi-simples are Morita equivalent to completions of path algebras. This fact motivated all the local-quiver technology that was developed here in Antwerp over the last decade (see my book if you want to know the details).

Also for 3-dimensional Calabi-Yau algebras it states that the completions at semi-simples are Morita equivalent to completions of quotients of path algebras by the relations coming from a superpotential (aka a necklace) by taking partial noncommutative derivatives. Here the essential ingredient is that $Ext^2_A(M,M)^{\ast} \simeq Ext^1_A(M,M) $ in this case.

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time for selfcriticism

The
problem with criticizing others is that you have to apply the same
standards to your own work. So, as of this afternoon, I do agree with
all those who said so before : my book is completely unreadable and
should either be dumped or entirely rewritten!

Here’s what happened :
Last week I did receive the contract to publish _noncommutative
geometry@n_ in a reputable series. One tiny point though, the editors
felt that the title was somewhat awkward and would stand out with
respect to the other books in the series, so they proposed as an
alternative title _Noncommutative Geometry_. A tall order, I thought,
but then, if others are publishing books with such a title why
shouldn’t I do the same?

The later chapters are quite general, anyway,
and if I would just spice them up a little adding recent material it
might even improve the book. So, rewriting two chapters and perhaps
adding another “motivational chapter” aimed at physicists… should
be doable in a month, or two at the latest which would fit in nicely
with the date the final manuscript is due.

This week, I got myself once
again in writing mode : painfully drafting new sections at a pace of 5
to 6 pages a day. Everything was going well. Today I wanted to finish
the section on the “one quiver to rule them all”-trick and was
already mentally planning the next section in which I would give details
for groups like $PSL_2(\mathbb{Z}) $ and $GL_2(\mathbb{Z}) $, all I
needed was to type in a version of the proof of the last proposition.

The proof uses a standard argument, which clearly should be in the book
so I had to give the correct reference and started browsing through the
print-out of the latest version (about 600 pages long..) but… _I
could not find it!???_ And, it was not just some minor technical lemma,
but a result which is crucial to the book’s message (for the few who
want to know, the result is the construction and properties of the local
quiver at a semi-simple representation of a Quillen-smooth algebra). Of
course, there is a much more general result contained in the book, but
you have to be me (or have to be drilled by me) to see the connection…
Not good at all! I’d better sleep on this before taking further
steps

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coalgebras and non-geometry 2

Last time we
have seen that the _coalgebra of distributions_ of an affine smooth
variety is the direct sum (over all points) of the dual to the etale
local algebras which are all of the form $\mathbb{C}[[
x_1,\ldots,x_d ]] $ where $d $ is the dimension of the
variety. Generalizing this to _non-commutative_ manifolds, the first
questions are : “What is the analogon of the power-series algebra?” and
do all ‘points’ of our non-commutative manifold do have such local
algebras? Surely, we no longer expect the variables to commute, so a
non-commutative version of the power series algebra should be
$\mathbb{C} \langle \langle x_1,\ldots,x_d \rangle \rangle $,
the ring of formal power series in non-commuting variables. However,
there is still another way to add non-commutativity and that is to go
from an algebra to matrices over the algebra. So, in all we would expect
to be our _local algebras_ at points of our non-commutative manifold to
be isomorphic to $M_n(\mathbb{C} \langle \langle x_1,\ldots,x_d
\rangle \rangle) $ As to the second question : _qurves_ (that is,
the coordinate rings of non-commutative manifolds) do have such algebras
as local rings provided we take as the ‘points’ of the non-commutative
variety the set of all _simple_ finite dimensional representations of
the qurve. This is a consequence of the _tubular neighborhood theorem_
due to [Cuntz](http://wwwmath.uni-muenster.de/u/cuntz/cuntz.html) and
[Quillen](http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Quillen.html). In more details : If A is a qurve, then a simple
$n $-dimensional representation corresponds to an epimorphism
$\pi~:~A \rightarrow S = M_n(\mathbb{C}) $ and if we take
$\mathfrak{m}=Ker(\pi) $, then
$M=\mathfrak{m}/\mathfrak{m}^2 $ is an $S $-bimodule and
the $\mathfrak{m} $-adic completion of A is isomorphic to the
completed tensor-algebra $\hat{T}_S(M) \simeq M_n(\mathbb{C}
\langle \langle x_1,\ldots,x_d \rangle \rangle) $ In contrast with
the commutative case however where the dimension remains constant over
all points, here the numbers n and d can change from simple to simple.
For n this is clear as it gives the dimension of the simple
representation, but also d changes (it is the local dimension of the
variety classifying simple representations of the same dimension). Here
an easy example : Consider the skew group algebra $A =
\mathbb{C}[x] \star C_2 $ with the action given by sending $x
\mapsto -x $. Then A is a qurve and its center is
$\mathbb{C}[y] $ with $y=x^2 $. Over any point $y
\not= 0 $ there is a unique simple 2-dimensional representation of A
giving the local algebra $M_2(\mathbb{C}[[y]]) $. If
$y=0 $ the situation is more complicated as the local structure
of A is given by the algebra $\begin{bmatrix} \mathbb{C}[[y]] &
\mathbb{C}[[y]] \\ (y) & \mathbb{C}[[y]] \end{bmatrix} $ So, over
this point there are precisely 2 one-dimensional simple representations
corresponding to the maximal ideals $\mathfrak{m}_1 =
\begin{bmatrix} (y) & \mathbb{C}[[y]] \\ (y) & \mathbb{C}[[y]]
\end{bmatrix}~\qquad \text{and}~\qquad \mathfrak{m}_2 = \begin{bmatrix}
\mathbb{C}[[y]] & \mathbb{C}[[y]] \\ (y) & (y) \end{bmatrix} $ and
both ideals are idempotent, that is $\mathfrak{m}_i^2 =
\mathfrak{m}_i $ whence the corresponding bimodule $M_i =
0 $ so the local algebra in either of these two points is just
$\mathbb{C} $. Ok, so the comleted local algebra at each point
is of the form $M_n(\mathbb{C}\langle \langle x_1,\ldots,x_d \rangle
\rangle) $, but what is the corresponding dual coalgebra. Well,
$\mathbb{C} \langle \langle x_1,\ldots,x_d \rangle \rangle $ is
the algebra dual to the _cofree coalgebra_ on $V = \mathbb{C} x_1 +
\ldots + \mathbb{C}x_d $. As a vectorspace this is the
tensor-algebra $T(V) = \mathbb{C} \langle x_1,\ldots,x_d
\rangle $ with the coalgebra structure induced by the bialgebra
structure defined by taking all varaibales to be primitives, that is
$\Delta(x_i) = x_i \otimes 1 + 1 \otimes x_i $. That is, the
coproduct on a monomial gives all different expressions $m_1 \otimes
m_2 $ such that $m_1m_2 = m $. For example,
$\Delta(x_1x_2) = x_1x_2 \otimes 1 + x_1 \otimes x_2 + 1 \otimes
x_1x_2 $. On the other hand, the dual coalgebra of
$M_n(\mathbb{C}) $ is the _matrix coalgebra_ which is the
$n^2 $-dimensional vectorspace $\mathbb{C}e_{11} + \ldots +
\mathbb{C}e_{nn} $ with comultiplication $\Delta(e_{ij}) =
\sum_k e_{ik} \otimes e_{kj} $ The coalgebra corresponding to the
local algebra $M_n(\mathbb{C}\langle \langle x_1,\ldots,x_d \rangle
\rangle) $ is then the tensor-coalgebra of the matrix coalgebra and
the cofree coalgebra. Having obtained the coalgebra at each point
(=simple representation) of our noncommutative manifold one might think
that the _coalgebra of non-commutative distributions_ should be the
direct sum of all this coalgebras, summed over all points, as in the
commutative case. But then we would forget about a major difference
between the commutative and the non-commutative world : distinct simples
can have non-trivial extensions! The mental picture one might have
about simples having non-trivial extensions is that these points lie
‘infinitesimally close’ together. In the $\mathbb{C}[x] \star
C_2 $ example above, the two one-dimensional simples have
non-trivial extensions so they should be thought of as a cluster of two
infinitesimally close points corresponding to the point $y=0 $
(that is, this commutative points splits into two non-commutative
points). Btw. this is the reason why non-commutative algebras can be
used to resolve commutative singularities (excessive tangents can be
split over several non-commutative points). While this is still pretty
harmless when the algebra is finite over its center (as in the above
example where only the two one-dimensionals have extensions), the
situation becomes weird over general qurves as ‘usually’ distinct
simples have non-trivial extensions. For example, for the free algebra
$\mathbb{C}\langle x,y \rangle $ this is true for all simples…
So, if we want to continue using this image of points lying closely
together this immediately means that non-commutative ‘affine’ manifolds
behave like compact ones (in fact, it turns out to be pretty difficult
to ‘glue’ together qurves into ‘bigger’ non-commutative manifolds, apart
from the quiver examples of [this old
paper](http://www.arxiv.org/abs/math.AG/9907136)). So, how to bring
this new information into our coalgebra of distributions? Well, let’s
repeat the previous argument not with just one point but with a set of
finitely many points. Then we have a _semi-simple algebra_ quotient
$\pi~:~A \rightarrow S = M_{n_1}(\mathbb{C}) \oplus \ldots \oplus
M_{n_k}(\mathb{C}) $ and taking again
$\mathfrak{m}=Ker(\pi) $ and
$M=\mathfrak{m}/\mathfrak{m}^2 $, then $M $ is again an
S-bimodule. Now, any S-bimodule can be encoded into a _quiver_ Q on k
points, the number of arrows from vertex i to vertex j being the number
of components in M of the form $M_{n_i \times
n_j}(\mathbb{C}) $. Again, it follows from the tubular neighborhood
theorem that the $\mathfrak{m} $-adic completion of A is
isomorphic to the completion of an algebra Morita equivalent to the
_path algebra_ $\mathbb{C} Q $ (being the tensor algebra
$T_S(M) $). As all the local algebras of the points are
quotients of this quiver-like completion, on the coalgebra level our
local coalgebras will be sub coalgebras of the coalgebra which is
co-Morita equivalent (and believe it or not but coalgebraists have a
name for this : _Takeuchi equivalence_) to the _quiver coalgebra_ which
is the vectorspace of the path algebra $\mathbb{C} Q $ with
multiplication induced by making all arrows from i to j skew-primitives,
that is, $\Delta(a) = e_i \otimes a + a \otimes e_j $ where the
$e_i $ are group-likes corresponding to the vertices. If all of
ths is a bit too much co to take in at once, I suggest the paper by Bill
Chin [A brief introduction to coalgebra representation
theory](http://condor.depaul.edu/~wchin/crt.pdf#search=%22%22A%20brief%20introduction%20to%20coalgebra%20representation%20theory%22%22). The
_coalgebra of noncommutative distributions_ we are after at is now the
union of all these Takeuchi-equivalent quiver coalgebras. In easy
examples such as the $\mathbb{C}[x] \star C_2 $-example this
coalgebra is still pretty small (the sum of the local coalgebras
corresponding to the local algebras $M_2(\mathbb{C}[[x]]) $
summed over all points $y \not= 0 $ summed with the quiver
coalgebra of the quiver $\xymatrix{\vtx{} \ar@/^/[rr] & & \vtx{}
\ar@/^/[ll]} $ In general though this is a huge object and we would
like to have a recipe to construct it from a manageable _blue-print_ and
that is what we will do next time.

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