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

## Comments