The previous post can be found here.
Pierre Gabriel invented a lot of new notation (see his book Representations of finite dimensional algebras for a rather extreme case) and is responsible for calling a directed graph a quiver. For example,

\xymatrix{\vtx{} \ar@/^/[rr] & & \vtx{} \ar@(u,ur) \ar@(d,dr) \ar@/^/[ll]}

is a quiver. Note than it is allowed to have multiple arrows between vertices, as well as loops in vertices. For us it will be important that a quiver Q depicts how to compute in a certain non-commutative algebra : the path algebra \C Q. If the quiver has k vertices and l arrows (including loops) then the path algebra \C Q is a subalgebra of the full k \times k matrix-algebra over the free algebra in l non-commuting variables

\C Q \subset M_k(\C \langle x_1,\hdots,x_l \rangle)

Under this map, a vertex v_i is mapped to the basis i-th diagonal matrix-idempotent and an arrow

\xymatrix{\vtx{v_i} \ar[rr]^{x_a} & & \vtx{v_j}}

is mapped to the matrix having all its entries zero except the tex[/tex]-entry which is equal to x_a. That is, in our main example

\xymatrix{\vtx{e} \ar@/^/[rr]^a & & \vtx{f} \ar@(u,ur)^x \ar@(d,dr)_y \ar@/^/[ll]^b}

the corresponding path algebra is the subalgebra of M_2(\C \langle a,b,x,y \rangle) generated by the matrices

e \mapsto \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} f \mapsto \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}

a \mapsto \begin{bmatrix} 0 & 0 \\ a & 0 \end{bmatrix} b \mapsto \begin{bmatrix} 0 & b \\ 0 & 0 \end{bmatrix}

x \mapsto \begin{bmatrix} 0 & 0 \\ 0 & x \end{bmatrix} y \mapsto \begin{bmatrix} 0 & 0 \\ 0 & y \end{bmatrix}

The name \’path algebra\’ comes from the fact that the subspace of \C Q at the tex[/tex]-place is the vectorspace spanned by all paths in the quiver starting at vertex v_i and ending in vertex v_j. For an easier and concrete example of a path algebra. consider the quiver

\xymatrix{\vtx{e} \ar[rr]^a & & \vtx{f} \ar@(ur,dr)^x}

and verify that in this case, the path algebra is just

\C Q = \begin{bmatrix} \C & 0 \\ \C[x]a & \C[x] \end{bmatrix}

Observe that we write and read paths in the quiver from right to left. The reason for this strange convention is that later we will be interested in left-modules rather than right-modules. Right-minder people can go for the more natural left to right convention for writing paths.
Why are path algebras of quivers of interest in non-commutative geometry? Well, to begin they are examples of formally smooth algebras (some say quasi-free algebras, I just call them qurves). These algebras were introduced and studied by Joachim Cuntz and Daniel Quillen and they are precisely the algebras allowing a good theory of non-commutative differential forms.
So you should think of formally smooth algebras as being non-commutative manifolds and under this analogy path algebras of quivers correspond to affine spaces. That is, one expects path algebras of quivers to turn up in two instances : (1) given a non-commutative manifold (aka formally smooth algebra) it must be \’embedded\’ in some non-commutative affine space (aka path algebra of a quiver) and (2) given a non-commutative manifold, the \’tangent spaces\’ should be determined by path algebras of quivers.
The first fact is easy enough to prove, every affine \C-algebra is an epimorphic image of a free algebra in say l generators, which is just the path algebra of the bouquet quiver having l loops

\xymatrix{\vtx{} \ar@(dl,l)^{x_1} \ar@(l,ul)^{x_2} \ar@(ur,r)^{x_i} \ar@(r,dr)^{x_l}}

The second statement requires more work. For a first attempt to clarify this you can consult my preprint Qurves and quivers but I\’ll come back to this in another post. For now, just take my word for it : if formally smooth algebras are the non-commutative analogon of manifolds then path algebras of quivers are the non-commutative version of affine spaces!

arxiv, Cuntz, differential, geometry, non-commutative, Quillen, quivers, representations