[Last
time][1] we saw that the algebra of relative
differential forms and equipped with the Fedosov product is again the
path algebra of a quiver obtained by doubling up the arrows
of . In our basic example the algebra map is clarified by the following picture of
(which
generalizes in the obvious way to arbitrary quivers). But what about the
other direction ? There are two
embeddings defined by and giving maps
Using these maps, the isomorphism is determined by In particular, gives the
natural embedding (with the ordinary multiplication on differential
forms) of functions as degree zero
differential forms. However, is no longer an algebra map for the
Fedosov product on as . In Cuntz-Quillen terminology, is
the _curvature_ of the based linear map . I\โd better define
this a bit more formal for any algebra and then say what is special
for formally smooth algebras (non-commutative manifolds). If are
-algebras, then a -linear map is said to be a _based linear map_ if .
The _curvature_ of measures the obstruction to being an algebra
map, that is and
the curvature is said to be _nilpotent_ if there is an integer such
that all possible products For any algebra there is a universal algebra
turning based linear maps into algebra maps. That is, there is a
fixed based linear map such that for every based
linear map there is an algebra map making the diagram commute In fact, Cuntz and Quillen show that the algebra of even differential forms
equipped with the Fedosov product and that is the natural inclusion
of as degree zero forms (as above). Recall that is said to be
_formally smooth_ if every -algebra map where
is a nilpotent ideal, can be lifted to an algebra morphism . We can always lift as a based linear map, say
and because is nilpotent, the curvature of
is also nilpotent. To get a _uniform_ way to construct algebra lifts
modulo nilpotent ideals it would therefore suffice for a formally smooth
algebra to have an _algebra map_ where
is the -adic completion of for the
ideal which is the kernel of the algebra map corresponding to the based linear map . Indeed, there is an algebra map
determined by and hence also an algebra map and composing this with the (yet to be constructed)
algebra map this would give the required lift
. In order to construct the algebra map (say in the case of path algebras of quivers) we
will need the Yang-Mills derivation and its associated flow.
[1]: https://lievenlb.local/index.php?p=354
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