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I heard from some guys working in noncommutative geometry talking about the idea that one can construct the noncommutative space from quivers. I feel it is rather interesting. However, I can not image how one can construct a non-affine scheme or space. For affine case, one might consider the path algebra of a quiver $Q$, say, $KQ$. It is a hereditary algebra(or quasi-free algebra, formally smooth algebra). How can one construct a non-affine space from quiver?

Is it possible to construct a noncommutative space using different interesting quivers as "local data"?

If it is possible, the algebraic geometry machine working on this space will be helpful to study the representations of quiver?


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It is possible to construct more general schemes (even 'projective' schemes) starting from quiver-representations.

The idea is to consider a stability structure and consider the corresponding moduli spaces of semi-stable quiver representations as introduced by Alastair King (Moduli of representations of finite dimensional algebras, Quat. J. Math. Oxford 45 (1994) 515-530.)

If one starts with a quiver without oriented cycles, these moduli spaces are projective and so it makes sense to take the collection of all these moduli spaces (or multiples of a fixed dimension vector compatible with the stability structure) as a noncommutative projective space. This point of view was advocated in 'Noncommutative compact manifolds constructed from quivers'.

More generally, allowing cycles, the collection of moduli spaces (or multiples of fixed dim vector) for all dim vectors compatible with the stabilitu structure can be viewed as a noncommutative scheme.

Here the main idea is that polynomial semi-invariants of quivers are known to be generated by so called determinental semi-invarinats (a result by Aidan Schofield and Michel Van den Bergh 'Semi-invariants of quivers for arbitrary dimension vectors'.

What this says is that these collections of moduli spaces have an affine cover determined by universal localizations of the path algebra. This point of view is expressed here.

EDIT : Here's an example to illustrate all of this : noncommutative projective n-space. Consider the quiver with 2 vertices and n+1 arrows, all from the first to the second vertex, Take stability structure (-1,1), then we consider only representations of dimension vector (m,m). A semi-stable representation is one given by the n+1 mxm matrices X0,...,Xn such that there are numbers ai with det(a0X0+a1X1+...+anXn) is non-zero. For example, consider the open piece where Xi is invertible, then one can use Xi to identify the two vertex-spaces and the other arrows now become loops in that common vertex. That is, this open piece gives all representations of the free algebra in n variables. Hence, we can cover the moduli space by affine pieces, all iso to the reps of a one-vertex quiver with n loops.

Clearly one can generalize this procedure to glue certain representations of two very different quivers. Formally invert one arrow in your quiver, this comes down to taking a stability structure with 0's at all other vertices and (-1,1) at start and end-point of your arrow. Then the 'Zariski open piece' of repQ consists the representations of a new quiver Q' in which one identifies the two vertices, and turns all other arrows between these two vertices into loops at the new vertex. In this way to quivers can be 'glued' to form more general schemes provided they are isomorphic after doing this 'invert one arrow' trick on each of them a finite number of times.

Universal localizations corresponding to 'determinental semi-invariants' are a generalization of this idea, allowing paths (or even collections of linear combinations of paths) to be 'inverted'. (End EDIT)

A recent incarnation of this strain of ideas (restricting to stable rather than semi-stable representations) is taken by Markus Reineke in his study of 'noncommutative Hilbert schemes' which have been used by Markus recently in 'Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants'.

As to the local structure of these noncommutative schemes. Their 'local rings' are indeed again path algebras of 'local quivers' which can be determined from the stable components of the semi-stable representation. The details are here.

If you want to extend all of this to general formally smooth algebras (rather than just path algebras) a place to start is with 'One quiver to rule them all'.

Finally, Ive tried to collect all of this in a book as Daniel mentioned in his answer (one might replace 'blablabla' by 'noncommutative geometry'...). The latest version of it can be downloaded here.

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Excellent! $ $ – Victor Protsak May 17 '10 at 19:12

As far as I know a "non-commutative space" is for many people just an $A_\infty$-category. Every scheme determines a non-commutative space: its derived category of coherent sheaves (with an $A_\infty$ enrichment).

Also to an (non-commuative) algebra $A$ you can associate a non-commutative space, the derived category of modules over $A$.

In some cases these two types of non-commutative space are isomorphic. The classical example (due to Belinson) is projective space whose derived category is equivalent to the derived category of modules over $End(O \oplus O(1) \oplus \dots O \oplus O(n) )$.

This algebra has a description as path algebra over a quiver with relations. E.g. for $P^1$ you get the Kronecker quiver (two vertices two arrows in the same direction).

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I think you should start looking at the webpage of Lieven Le Bruyn. He most definitely has some notes that will quell your thirst in this respect. For one he has written a book that is freely available on his webpage ("Noncommutative Geometry and Cayley-smooth Orders"). Another good person whose papers and webpage it's reasonable to pay a visit to if you're interested in this area is Markus Reineke.

Victor Ginzburg's notes on Noncommutative geometry arXiv:math/0506603 can maybe also be of interest even though I don't know if there are any quivers lurking there.

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