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 $\mathbb{C}[[x_1,\dots ,x_d]]$ 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
$\mathbb{C}⟨⟨x_1,\dots ,x_d⟩⟩$,
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 $M_n(\mathbb{C}⟨⟨x_1,\dots ,x_d⟩⟩)$ 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
$\pi :A\to S=M_n(\mathbb{C})$ 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 ${\hat{T}}_S(M)\simeq M_n(\mathbb{C}⟨⟨x_1,\dots ,x_d⟩⟩)$ 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 $y\ne 0$ 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 $\left[\begin{array}{cc}\mathbb{C}[[y]]& \mathbb{C}[[y]]\\ (y)& \mathbb{C}[[y]]\end{array}\right]$ So, over
this point there are precisely 2 one-dimensional simple representations
corresponding to the maximal ideals ${\mathfrak{m}}_1=\left[\begin{array}{cc}(y)& \mathbb{C}[[y]]\\ (y)& \mathbb{C}[[y]]\end{array}\right]\text{and}{\mathfrak{m}}_2=\left[\begin{array}{cc}\mathbb{C}[[y]]& \mathbb{C}[[y]]\\ (y)& (y)\end{array}\right]$ 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 $M_n(\mathbb{C}⟨⟨x_1,\dots ,x_d⟩⟩)$, but what is the corresponding dual coalgebra. Well,
$\mathbb{C}⟨⟨x_1,\dots ,x_d⟩⟩$ is
the algebra dual to the _cofree coalgebra_ on $V=\mathbb{C}{x}_1+\dots +\mathbb{C}{x}_d$. As a vectorspace this is the
tensor-algebra $T(V)=\mathbb{C}⟨x_1,\dots ,x_d⟩$ with the coalgebra structure induced by the bialgebra
structure defined by taking all varaibales to be primitives, that is
$\mathrm{\Delta }(x_i)=x_i\otimes 1+1\otimes x_i$. That is, the
coproduct on a monomial gives all different expressions $m_1\otimes m_2$ such that $m_1{m}_2=m$. For example,
$\mathrm{\Delta }(x_1{x}_2)=x_1{x}_2\otimes 1+x_1\otimes x_2+1\otimes x_1{x}_2$. On the other hand, the dual coalgebra of
$M_n(\mathbb{C})$ is the _matrix coalgebra_ which is the
$n^2$-dimensional vectorspace $\mathbb{C}{e}_{11}+\dots +\mathbb{C}{e}_{nn}$ with comultiplication $\mathrm{\Delta }(e_{ij})=\sum _k{e}_{ik}\otimes e_{kj}$ The coalgebra corresponding to the
local algebra $M_n(\mathbb{C}⟨⟨x_1,\dots ,x_d⟩⟩)$ 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
$\mathbb{C}⟨x,y⟩$ 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
$\pi :A\to S=M_{n_1}(\mathbb{C})\oplus \dots \oplus M_{n_k}(\text{mathb}C)$ 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, $\mathrm{\Delta }(a)=e_i\otimes a+a\otimes e_j$ 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 $y\ne 0$ summed with the quiver
coalgebra of the quiver 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.