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 
.
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 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)](/latexrender/pictures/b5269514761f14ba09e7e551b2803b36.gif "~(\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 
arxiv, coalgebras, geometry, groups, Kontsevich, noncommutative, puzzle, quivers, representations, simples
No comments
Posted in geometry
Written on Mon, 11 September 2006 at 3:05 pm
Tags: arxiv, coalgebras, geometry, groups, Kontsevich, noncommutative, puzzle, quivers, representations, simples
If you liked this post, then consider subscribing to our full RSS feed.
Leave a Reply