The

previous post in this sequence was [(co)tangent bundles][1]. Let $A$ be

a $V$-algebra where $V = C \times \ldots \times C$ 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][2] 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^n_V~A = A \otimes_V

\overline{A} \otimes_V \ldots \otimes_V \overline{A}$ with $n$ factors

$\overline{A}$. To get the connection with usual differential forms let

us denote the tensor $a_0 \otimes a_1 \otimes \ldots \otimes a_n =

(a_0,a_1,\ldots,a_n) = a_0 da_1 \ldots da_n$ On $\Omega_V~A =

\oplus_n~\Omega^n_V~A$ one defines an algebra structure via the

multiplication $(a_0da_1 \ldots da_n)(a_{n+1}da_{n+2} \ldots da_k)$$=

\sum_{i=1}^n (-1)^{n-i} a_0da_1 \ldots d(a_ia_{i+1}) \ldots da_k$

$\Omega_V~A$ is a _differential graded algebra_ with differential $d :

\Omega^n_V~A \rightarrow \Omega^{n+1}_V~A$ defined by $d(a_0 da_1 \ldots

da_n) = da_0 da_1 \ldots da_n$ This may seem fairly abstract but in

case $A = C Q$ is a path algebra, then the bimodule $\Omega^n_V~A$ has a

$V$-generating set consisting of precisely the elements $p_0 dp_1

\ldots dp_n$ with all $p_i$ non-zero paths in $A$ and such that

$p_0p_1 \ldots p_n$ is also a non-zero path. One can put another

algebra multiplication on $\Omega_V~A$ 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 $\Omega_V~A$ (with Fedosov

product) such that the _associated graded algebra_ is $\Omega_V~A$ with

the usual product. One can visualize the Fedosov product easily in the

case of path algebras because $\Omega_V~C Q$ 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

$\xymatrix{\vtx{} \ar[rr]^u & & \vtx{} \ar@(ur,dr)^v} $ the

algebra of non-commutative differential forms equipped with the Fedosov

product is isomorphic to the path algebra of the quiver

$\xymatrix{\vtx{} \ar@/^/[rr]^{a=u+du} \ar@/_/[rr]_{b=u-du} & &

\vtx{} \ar@(u,ur)^{x=v+dv} \ar@(d,dr)_{y=v-dv}} $ with the

indicated identification of arrows with elements from $\Omega_V~C Q$.

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][3].

[1]: http://www.neverendingbooks.org/index.php?p=352

[2]: http://www.neverendingbooks.org/index.php?p=349

[3]: http://www.neverendingbooks.org/index.php?p=353

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