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.

# coalgebras and non-geometry 2

Published in featured