# Tag: Cuntz

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.

In this
series of posts I’ll try to make at least part of the recent
[Kontsevich-Soibelman paper](http://www.arxiv.org/abs/math.RA/0606241) a
bit more accessible to algebraists. In non-geometry, the algebras
corresponding to *smooth affine varieties* I’ll call **qurves** (note
that they are called **quasi-free algebras** by Cuntz & Quillen and
**formally smooth** by Kontsevich). By definition, a qurve in an affine
$\mathbb{C}$-algebra A having the lifting property for algebra
maps through nilpotent ideals (extending Grothendieck’s characterization
of smooth affine algebras in the commutative case). Examples of qurves
are : finite dimensional semi-simple algebras (for example, group
algebras $\mathbb{C} G$ of finite groups), coordinate rings of
smooth affine curves or a noncommutative mixture of both, skew-group
algebras $\mathbb{C}[X] \ast G$ whenever G is a finite group of
automorphisms of the affine curve X. These are Noetherian examples but
in general a qurve is quite far from being Noetherian. More typical
examples of qurves are : free algebras $\mathbb{C} \langle x_1,\ldots,x_k \rangle$ and path algebras of finite quivers
$~\mathbb{C} Q$. Recall that a finite quiver Q s just a
directed graph and its path algebra is the vectorspace spanned by all
directed paths in Q with multiplication induced by concatenation of
paths. Out of these building blocks one readily constructs more
involved qurves via universal algebra operations such as (amalgamated)
free products, universal localizations etc. In this way, the
groupalgebra of the modular group $SL_2(\mathbb{Z})$ (as well
as that of a congruence subgroup) is a qurve and one can mix groups with
finite groupactions on curves to get qurves like $(\mathbb{C}[X] \ast G) \ast_{\mathbb{C} H} \mathbb{C} M$ whenever H is a common
subgroup of the finite groups G and M. So we have a huge class of
qurve-examples obtained from mixing finite and arithmetic groups with
curves and quivers. Qurves can we used as *machines* generating
interesting $A_{\infty}$-categories. Let us start by recalling
some facts about finite closed subschemes of an affine smooth variety Y
in the commutative case. Let **fdcom** be the category of all finite
dimensional commutative $\mathbb{C}$-algebras with morphisms
being onto algebra morphisms, then the study of finite closed subschemes
of Y is essentially the study of the covariant functor **fdcom** –>
**sets** assigning to a f.d. commutative algebra S the set of all onto
algebra maps from $\mathbb{C}[Y]$ to S. S being a f.d.
commutative semilocal algebra is the direct sum of local factors $S \simeq S_1 \oplus \ldots \oplus S_k$ where each factor has a
unique maximal ideal (a unique point in Y). Hence, our study reduces to
f.d. commutative images with support in a fixed point p of Y. But all
such quotients are also quotients of the completion of the local ring of
Y at p which (because Y is a smooth variety, say of dimension n) is
isomorphic to formal power series
$~\mathbb{C}[[x_1,\ldots,x_n]]$. So the local question, at any
point p of Y, reduces to finding all settings
$\mathbb{C}[[x_1,\ldots,x_n]] \twoheadrightarrow S \twoheadrightarrow \mathbb{C}$ Now, we are going to do something
strange (at least to an algebraist), we’re going to take duals and
translate the above sequence into a coalgebra statement. Clearly, the
dual $S^{\ast}$ of any finite dimensional commutative algebra
is a finite dimensional cocommutative coalgebra. In particular
$\mathbb{C}^{\ast} \simeq \mathbb{C}$ where the
comultiplication makes 1 into a grouplike element, that is
$\Delta(1) = 1 \otimes 1$. As long as the (co)algebra is
finite dimensional this duality works as expected : onto maps correspond
to inclusions, an ideal corresponds to a sub-coalgebra a sub-algebra
corresponds to a co-ideal, so in particular a local commutative algebra
corresponds to an pointed irreducible cocommutative coalgebra (a
coalgebra is said to be irreducible if any two non-zero subcoalgebras
have non-zero intersection, it is called simple if it has no non-zero
proper subcoalgebras and is called pointed if all its simple
subcoalgebras are one-dimensional. But what about infinite dimensional
algebras such as formal power series? Well, here the trick is not to
take all dual functions but only those linear functions whose kernel
contains a cofinite ideal (which brings us back to the good finite
dimensional setting). If one takes only those good linear functionals,
the ‘fancy’-dual $A^o$of an algebra A is indeed a coalgebra. On
the other hand, the full-dual of a coalgebra is always an algebra. So,
between commutative algebras and cocommutative coalgebras we have a
duality by associating to an algebra its fancy-dual and to a coalgebra
its full-dual (all this is explained in full detail in chapter VI of
Moss Sweedler’s book ‘Hopf algebras’). So, we can dualize the above pair
of onto maps to get coalgebra inclusions $\mathbb{C} \subset S^{\ast} \subset U(\mathfrak{a})$ where the rightmost coalgebra is
the coalgebra structure on the enveloping algebra of the Abelian Lie
algebra of dimension n (in which all Lie-elements are primitive, that is
$\Delta(x) = x \otimes 1 + 1 \otimes x$ and indeed we have that
$U(\mathfrak{a})^{\ast} \simeq \mathbb{C}[[x_1,\ldots,x_n]]$.
We have translated our local problem to finding all f.d. subcoalgebras
(containing the unique simple) of the enveloping algebra. But what is
the point of this translation? Well, we are not interested in the local
problem, but in the global problem, so we somehow have to **sum over all
points**. Now, on the algebra level that is a problem because the sum of
all local power series rings over all points is no longer an algebra,
whereas the direct sum of all pointed irreducible coalgebras $~B_Y = \oplus_{p \in Y} U(\mathfrak{a}_p)$ is again a coalgebra! That
is, we have found a huge coalgebra (which we call the coalgebra of
‘distributions’ on Y) such that for every f.d. commutative algebra S we
have $Hom_{comm alg}(\mathbb{C}[Y],S) \simeq Hom_{cocomm coalg}(S^{\ast},B_Y)$ Can we get Y back from this coalgebra of
districutions? Well, in a way, the points of Y correspond to the
group-like elements, and if g is the group-like corresponding to a point
p, we can recover the tangent-space at p back as the g-primitive
elements of the coalgebra of distributions, that is the elements such
that $\Delta(x) = x \otimes g + g \otimes x$. Observe that in
this commutative case, there are no **skew-primitives**, that is
elements such that $\Delta(x) = x \otimes g + h \otimes x$ for
different group-likes g and h. This is the coalgebra translation of the
fact that a f.d. semilocal commutative algebra is the direct sum of
local components. This is something that will definitely change if we
try to extend the above to the case of qurves (to be continued).

Now
that my non-geometry
post
string-coffee-table post
which in turn is available through a
paper

The little
this : if you have a situation where a discrete group is acting with a
bad orbit-space (for example, $GL_2(\mathbb{Z})$ acting on the whole
complex-plane, rather than just the upper half plane) you can associate
to this a $C^*$-algebra and study invariants of it and interprete them
example is the one I mentioned and where the additional noncommutative
points (coming from the orbits on the real axis) seem to contain a lot
of modular information as clarified by work of Manin&Marcolli and
Zagier. Probably the best introduction into Connes-style
non-commutative geometry
from this perspective are the Lecture on
Arithmetic Noncommutative Geometry
by Matilde Marcolli. To
algebraists : this trick is very similar to looking at the
skew-group algebra $\mathbb{C}[x_1,\ldots,x_n] * G$ if
you want to study the _orbifold_ for a finite group action on affine
space. But as algebraist we have to stick to affine varieties and
polynomials so we can only deal with the case of a finite group,
analysts can be sloppier in their functions, so they can also do
something when the group is infinite.

By the way, the
skew-group algebra idea is also why non-commutative algebraic
geometry
enters string-theory via the link with orbifolds. The
easiest (and best understood) example is that of Kleinian singularities.
The best introduction to this idea is via the Representations
of quivers, preprojective algebras and deformations of quotient
singularities
notes by Bill Crawley-Boevey.

Artin-style non-commutative geometry aka
non-commutative projective geometry originated from the
work of Artin-Tate-Van den Bergh (in the west) and Odeskii-Feigin (in
the east) to understand Sklyanin algebras associated to elliptic curves
and automorphisms via ‘geometric’ objects such as point- (and
fat-point-) modules, line-modules and the like. An excellent survey
paper on low dimensional non-commutative projective geometry is Non-commutative curves and surfaces by Toby
Stafford and
Michel Van den Bergh
. The best introduction is the (also
neverending…) book-project Non-
commutative algebraic geometry
by Paul Smith who
maintains a
noncommutative geometry and algebra resource page
page (which is

Non-geometry
started with the seminal paper ‘Algebra extensions and
nonsingularity’, J. Amer. Math. Soc. 8 (1995), 251-289 by Joachim
Cuntz
and Daniel Quillen but which is not available online. An
online introduction is Noncommutative smooth
spaces
by Kontsevich and Rosenberg. Surely, different people have
different motivations to study non-geometry. I assume Cuntz got
interested because inductive limits of separable algebras are quasi-free
(aka formally smooth aka qurves). Kontsevich and Soibelman want to study
morphisms and deformations of $A_{\infty}$-categories as they explain in
their recent
paper
. My own motivation to be interested in non-geometry is the
hope that in the next decades one will discover new exciting connections
between finite groups, algebraic curves and arithmetic groups (monstrous
moonshine
being the first, and still not entirely understood,
instance of this). Part of the problem is that these three topics seem
to be quite different, yet by taking group-algebras of finite or
arithmetic groups and coordinate rings of affine smooth curves they all
turn out to be quasi-free algebras, so perhaps non-geometry is the
unifying theory behind these seemingly unrelated topics.