differential forms

The previous post in this sequence was (co)tangent bundles. Let $A$ be a $V$-algebra where $V = C \times \hdots \times C$ is the subalgebra generated by a complete set of orthogonal idempotents in $A$ (in case $A = C Q$ is a path algebra, $V$ will be the subalgebra generated by the vertex-idempotents, see the post on path algebras for more details). With $\overline{A}$ we denote the bimodule quotient
$\overline{A} = A/V$ Then, we can define the non-commutative (relative) differential n-forms to be $\Omega^nV~A = A \otimesV \overline{A} \otimesV \hdots \otimesV \overline{A}$ with $n$ factors $\overline{A}$. To get the connection with usual differential forms let us denote the tensor $a0 \otimes a1 \otimes \hdots \otimes an = (a0,a1,\hdots,an) = a0 da1 \hdots dan$ On $\OmegaV~A = \oplusn~\Omega^nV~A$ one defines an algebra structure via the multiplication $(a0da1 \hdots dan)(a{n+1}da{n+2} \hdots dak)$$= \sum{i=1}^n (-1)^{n-i} a0da1 \hdots d(aia{i+1}) \hdots dak$
$\OmegaV~A$ is a _differential graded algebra with differential $d : \Omega^nV~A \rightarrow \Omega^{n+1}V~A$ defined by $d(a0 da1 \hdots dan) = da0 da1 \hdots dan$ This may seem fairly abstract but in case $A = C Q$ is a path algebra, then the bimodule $\Omega^nV~A$ has a $V$-generating set consisting of precisely the elements $p0 dp1 \hdots dpn$ with all $pi$ non-zero paths in $A$ and such that $p0p1 \hdots pn$ is also a non-zero path. One can put another algebra multiplication on $\OmegaV~A$ 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 $\OmegaV~A$ (with Fedosov product) such that the _associated graded algebra is $\OmegaV~A$ with the usual product. One can visualize the Fedosov product easily in the case of path algebras because $\OmegaV~C Q$ 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 $\OmegaV~C Q$. Note however that we usually embed the algebra $C Q$ as the degree zero differential forms in $\Omega_V~C Q$ 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.

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