differential forms

The
previous post in this sequence was [(co)tangent bundles][1]. Let $A$ be
a $V$-algebra where $V=C\times \dots \times C$ is the subalgebra
generated by a complete set of orthogonal idempotents in $A$ (in case $A=CQ$ is a path algebra, $V$ will be the subalgebra generated by the
vertex-idempotents, see the post on [path algebras][2] for more
details). With $\bar{A}$ we denote the bimodule quotient
$\bar{A}=A/V$ Then, we can define the _non-commutative
(relative) differential n-forms_ to be ${\mathrm{\Omega }}_V^{n}A=A{\otimes }_V\bar{A}{\otimes }_V\dots {\otimes }_V\bar{A}$ with $n$ factors
$\bar{A}$. To get the connection with usual differential forms let
us denote the tensor $a_0\otimes a_1\otimes \dots \otimes a_n=(a_0,a_1,\dots ,a_n)=a_{0}d{a}_1\dots d{a}_n$ On ${\mathrm{\Omega }}_{V}A={\oplus }_n{\mathrm{\Omega }}_V^{n}A$ one defines an algebra structure via the
multiplication $(a_{0}d{a}_1\dots d{a}_n)(a_{n+1}d{a}_{n+2}\dots d{a}_k)$$=\sum _{i=1}^n(-1{)}^{n-i}{a}_{0}d{a}_1\dots d(a_i{a}_{i+1})\dots d{a}_k$
${\mathrm{\Omega }}_{V}A$ is a _differential graded algebra_ with differential $d:{\mathrm{\Omega }}_V^{n}A\to {\mathrm{\Omega }}_V^{n+1}A$ defined by $d(a_{0}d{a}_1\dots d{a}_n)=d{a}_{0}d{a}_1\dots d{a}_n$ This may seem fairly abstract but in
case $A=CQ$ is a path algebra, then the bimodule ${\mathrm{\Omega }}_V^{n}A$ has a
$V$-generating set consisting of precisely the elements $p_{0}d{p}_1\dots d{p}_n$ with all $p_i$ non-zero paths in $A$ and such that
$p_0{p}_1\dots p_n$ is also a non-zero path. One can put another
algebra multiplication on ${\mathrm{\Omega }}_{V}A$ which Cuntz and Quillen call the
_Fedosov product_ defined for an $n$-form $\omega$ and a form $\mu$ by
$\omega Circ\mu =\omega \mu -(-1{)}^{n}d\omega d\mu$ There is an
important relation between the two structures, the degree of a
differential form puts a filtration on ${\mathrm{\Omega }}_{V}A$ (with Fedosov
product) such that the _associated graded algebra_ is ${\mathrm{\Omega }}_{V}A$ with
the usual product. One can visualize the Fedosov product easily in the
case of path algebras because ${\mathrm{\Omega }}_{V}CQ$ with the Fedosov product is
again the path algebra of the quiver obtained by doubling up all the
arrows of $Q$. In our basic example when $Q$ is the quiver
the
algebra of non-commutative differential forms equipped with the Fedosov
product is isomorphic to the path algebra of the quiver
with the
indicated identification of arrows with elements from ${\mathrm{\Omega }}_{V}CQ$.
Note however that we usually embed the algebra $CQ$ as the degree zero
differential forms in ${\mathrm{\Omega }}_{V}CQ$ with the usual multiplication and
that this embedding is no longer an algebra map (but a based linear map)
for the Fedosov product. For this reason, Cuntz and Quillen invent a
Yang-Mills type argument to “flow” this linear map to an algebra
embedding, but to motivate this we will have to say some things about
[curvatures][3].

[1]: https://lievenlb.local/index.php?p=352
[2]: https://lievenlb.local/index.php?p=349
[3]: https://lievenlb.local/index.php?p=353