Here a list of saved pdf-files of previous NeverEndingBooks-posts on geometry in reverse chronological order.

Leave a Comment# Tag: coalgebras

Last

time we saw that the _coalgebra of distributions_ of a

noncommutative manifold can be described as a coalgebra

Takeuchi-equivalent to the path coalgebra of a huge quiver. This

infinite quiver has as its vertices the isomorphism classes of finite

dimensional simple representations of the qurve A (the coordinate ring

of the noncommutative manifold) and there are as many directed arrows

between the vertices corresponding to the simples S and T as is the

dimension of $Ext^1_A(S,T) $.

The fact that this

coalgebra of distributions is equivalent to the path coalgebra of

_some_ quiver is in the Kontsevich-Soibelman

paper though it would have been nice if they had given reference for

this fact to the paper Wedge Products and

Cotensor Coalgebras in Monoidal Categories by Ardizzoni or to

previous work by P. Jara, D. Llena, L. Merino and D. Stefan,

“Hereditary and formally smooth coalgebras”, Algebr.

Represent. Theory 8 (2005), 363-374. In those papers it is shown that a

coalgebra with coseparable coradical is hereditary if and only if it

is formally smooth if and only if it is a cotensor coalgebra of some

bicomodule.

At first this looks just like the dual version of

the classical result that a finite dimensional hereditary algebra is

Morita equivalent to the path algebra of a quiver (which is indeed what

the proof does) but again the condition that the coradical is

coseparable does not require the coradical to be finite dimensional…

In our case, the coradical is indeed coseparable being the direct sum

over all matrix coalgebras corresponding to the simple representations.

Hence, we can again recover the _points_ of our noncommutative manifold

from the direct summands of the coradical. Fortunately, one can

compute this huge coalgebra of distributions from a small quiver, the

_one quiver to rule them all_, but as I’ve been babbling about all of

this here [numerous

times](http://www.neverendingbooks.org/?s=one+quiver) I’ll let the

interested find out for themselves how you use it (a) to get at the

isoclasses of all simples (hint : morally they are the smooth points of

the quotient varieties of n-dimensional representations and enough tools

have been developed recently to spot some fake simples, that is smooth

proper semi-simple points) and (b) to compute the _ragball_, that is the

huge quiver with vertex set the simples and arows as described

above. Over the years I’ve calculated several one-quivers for a

variety of qurves (such as amalgamated free products of finite groups

and smooth curves). If you are in for a puzzle, try to determine it for

the qurve $~(\mathbb{C}[x] \ast C_2) \ast_{\mathbb{C}

C_2} \mathbb{C} PSL_2(\mathbb{Z}) \ast_{\mathbb{C} C_3}

(\mathbb{C}[x] \ast C_3) $ The answer is a mysterious

hexagon

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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.