on September 14, 2004 by lieven in geometry, Comments (0)
differential forms
non-commutative geometry
- Brauer’s forgotten group
- connected component coalgebra
- Galois and the Brauer group
- a noncommutative Grothendieck topology
- noncommutative geometry
- noncommutative geometry 2
- projects in noncommutative geometry
- points and lines
- more noncommutative manifolds
- the necklace Lie bialgebra
- the one quiver for GL(2,Z)
- representation spaces
- quiver representations
- moduli spaces
- cotangent bundles
- differential forms
- curvatures
- Brauer-Severi varieties
- smooth Brauer-Severis
- hyper-resolutions
- a cosmic Galois group
- double Poisson algebras
- A for aggregates
- B for bricks
- necklaces (again)
- seen this quiver?
- why nag? (2)
- why nag? (3)
- sexing up curves
- the Klein stack
- Alain Connes on everything
- noncommutative topology (1)
- a noncommutative topology 2
- noncommutative topology (3)
- noncommutative topology (4)
- non-geometry
- non-(commutative) geometry
- noncommutative Fourier transform
- noncommutative bookmarks
- noncommutative geometry : a medieval science?
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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