Skip to content →

path algebras

The previous post can be found [here][1].
Pierre Gabriel invented a lot of new notation (see his book [Representations of finite dimensional algebras][2] for a rather extreme case) and is responsible for calling a directed graph a quiver. For example,

Misplaced &

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 CQ. If the quiver has k vertices and l arrows (including loops) then the path algebra CQ is a subalgebra of the full k×k matrix-algebra over the free algebra in l non-commuting variables

CQMk(Cx1,,xl)

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

Misplaced &

is mapped to the matrix having all its entries zero except the (j,i)-entry which is equal to xa. That is, in our main example

Misplaced &

the corresponding path algebra is the subalgebra of M2(Ca,b,x,y) generated by the matrices

e[1000] f[0001]

a[00a0] b[0b00]

x[000x] y[000y]

The name \’path algebra\’ comes from the fact that the subspace of CQ at the (j,i)-place is the vectorspace spanned by all paths in the quiver starting at vertex vi and ending in vertex vj. For an easier and concrete example of a path algebra. consider the quiver

Misplaced &

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

CQ=[C0C[x]aC[x]]

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)x1\ar@(l,ul)x2\ar@(ur,r)xi\ar@(r,dr)xl

The second statement requires more work. For a first attempt to clarify this you can consult my preprint [Qurves and quivers][3] 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!

[1]: https://lievenlb.local/index.php?p=71
[2]: http://www.booxtra.de/verteiler.asp?site=artikel.asp&wea=1070000&sh=homehome&artikelnummer=000000689724
[3]: http://www.arxiv.org/abs/math.RA/0406618

Published in featured

Comments

Leave a Reply

Your email address will not be published. Required fields are marked *