In what
way is a formally smooth algebra a _machine_ producing families of
manifolds? Consider the special case of the path algebra $\mathbb{C} Q$ of a
quiver and recall that an $n$-dimensional representation is an algebra
map $\mathbb{C} Q \rightarrow^{\phi} M_n(\mathbb{C})$ or, equivalently, an
$n$-dimensional left $\mathbb{C} Q$-module $\mathbb{C}^n_{\phi}$ with the action
determined by the rule $a.v = \phi(a) v~\forall v \in \mathbb{C}^n_{\phi},
\forall a \in \mathbb{C} Q$ If the $e_i~1 \leq i \leq k$ are the idempotents
in $\mathbb{C} Q$ corresponding to the vertices (see this [post][1]) then we get
a direct sum decomposition $\mathbb{C}^n_{\phi} = \phi(e_1)\mathbb{C}^n_{\phi} \oplus
\ldots \oplus \phi(e_k)\mathbb{C}^n_{\phi}$ and so every $n$-dimensional
representation does determine a _dimension vector_ $\alpha =
(a_1,\ldots,a_k)~\text{with}~a_i = dim_{\mathbb{C}} V_i = dim_{\mathbb{C}}
\phi(e_i)\mathbb{C}^n_{\phi}$ with $ | \alpha | = \sum_i a_i = n$. Further,
for every arrow $\xymatrix{\vtx{e_i} \ar[rr]^a & &
\vtx{e_j}} $ we have (because $e_j.a.e_i = a$ that $\phi(a)$
defines a linear map $\phi(a)~:~V_i \rightarrow V_j$ (that was the
whole point of writing paths in the quiver from right to left so that a
representation is determined by its _vertex spaces_ $V_i$ and as many
linear maps between them as there are arrows). Fixing vectorspace bases
in the vertex-spaces one observes that the space of all
$\alpha$-dimensional representations of the quiver is just an affine
space $\mathbf{rep}_{\alpha}~Q = \oplus_a~M_{a_j \times a_i}(\mathbb{C})$ and
base-change in the vertex-spaces does determine the action of the
_base-change group_ $GL(\alpha) = GL_{a_1} \times \ldots \times
GL_{a_k}$ on this space. Finally, as all this started out with fixing
a bases in $\mathbb{C}^n_{\phi}$ we get the affine variety of all
$n$-dimensional representations by bringing in the base-change
$GL_n$-action (by conjugation) and have $\mathbf{rep}_n~\mathbb{C} Q =
\bigsqcup_{| \alpha | = n} GL_n \times^{GL(\alpha)}
\mathbf{rep}_{\alpha}~Q$ and in this decomposition the connected
components are no longer just affine spaces with a groupaction but
essentially equal to them as there is a natural one-to-one
correspondence between $GL_n$-orbits in the fiber-bundle $GL_n
\times^{GL(\alpha)} \mathbf{rep}_{\alpha}~Q$ and $GL(\alpha)$-orbits in the
affine space $\mathbf{rep}_{\alpha}~Q$. In our main example
$\xymatrix{\vtx{e} \ar@/^/[rr]^a & & \vtx{f} \ar@(u,ur)^x
\ar@(d,dr)_y \ar@/^/[ll]^b} $ an $n$-dimensional representation
determines vertex-spaces $V = \phi(e) \mathbb{C}^n_{\phi}$ and $W = \phi(f)
\mathbb{C}^n_{\phi}$ of dimensions $p,q~\text{with}~p+q = n$. The arrows
determine linear maps between these spaces $\xymatrix{V
\ar@/^/[rr]^{\phi(a)} & & W \ar@(u,ur)^{\phi(x)} \ar@(d,dr)_{\phi(y)}
\ar@/^/[ll]^{\phi(b)}} $ and if we fix a set of bases in these two
vertex-spaces, we can represent these maps by matrices
$\xymatrix{\mathbb{C}^p \ar@/^/[rr]^{A} & & \mathbb{C}^q \ar@(u,ur)^{X}
\ar@(d,dr)_{Y} \ar@/^/[ll]^{B}} $ which can be considered as block
$n \times n$ matrices $a \mapsto \begin{bmatrix} 0 & 0 \\ A & 0
\end{bmatrix}~b \mapsto \begin{bmatrix} 0 & B \\ 0 & 0 \end{bmatrix}$
$x \mapsto \begin{bmatrix} 0 & 0 \\ 0 & X \end{bmatrix}~y \mapsto
\begin{bmatrix} 0 & 0 \\ 0 & Y \end{bmatrix}$ The basechange group
$GL(\alpha) = GL_p \times GL_q$ is the diagonal subgroup of $GL_n$
$GL(\alpha) = \begin{bmatrix} GL_p & 0 \\ 0 & GL_q \end{bmatrix}$ and
acts on the representation space $\mathbf{rep}_{\alpha}~Q = M_{q \times
p}(\mathbb{C}) \oplus M_{p \times q}(\mathbb{C}) \oplus M_q(\mathbb{C}) \oplus M_q(\mathbb{C})$
(embedded as block-matrices in $M_n(\mathbb{C})^{\oplus 4}$ as above) by
simultaneous conjugation. More generally, if $A$ is a formally smooth
algebra, then all its representation varieties $\mathbf{rep}_n~A$ are
affine smooth varieties equipped with a $GL_n$-action. This follows more
or less immediately from the definition and [Grothendieck][2]\’s
characterization of commutative regular algebras. For the record, an
algebra $A$ is said to be _formally smooth_ if for every algebra map $A
\rightarrow B/I$ with $I$ a nilpotent ideal of $B$ there exists a lift
$A \rightarrow B$. The path algebra of a quiver is formally smooth
because for every map $\phi~:~\mathbb{C} Q \rightarrow B/I$ the images of the
vertex-idempotents form an orthogonal set of idempotents which is known
to lift modulo nilpotent ideals and call this lift $\psi$. But then also
every arrow lifts as we can send it to an arbitrary element of
$\psi(e_j)\pi^{-1}(\phi(a))\psi(e_i)$. In case $A$ is commutative and
$B$ is allowed to run over all commutative algebras, then by
Grothendieck\’s criterium $A$ is a commutative regular algebra. This
also clarifies why so few commutative regular algebras are formally
smooth : being formally smooth is a vastly more restrictive property as
the lifting property extends to all algebras $B$ and whenever the
dimension of the commutative variety is at least two one can think of
maps from its coordinate ring to the commutative quotient of a
non-commutative ring by a nilpotent ideal which do not lift (for an
example, see for example [this preprint][3]). The aim of
non-commutative algebraic geometry is to study _families_ of manifolds
$\mathbf{rep}_n~A$ associated to the formally-smooth algebra $A$. [1]:
http://www.matrix.ua.ac.be/wp-trackback.php/10 [2]:
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.
html [3]: http://www.arxiv.org/abs/math.AG/9904171
Category: featured
The
previous part of this sequence was [quiver representations][1]. When $A$
is a formally smooth algebra, we have an infinite family of smooth
affine varieties $\mathbf{rep}_n~A$, the varieties of finite dimensional
representations. On $\mathbf{rep}_n~A$ there is a basechange action of
$GL_n$ and we are really interested in _isomorphism classes_ of
representations, that is, orbits under this action. Mind you, an orbit
space does not always exist due to the erxistence of non-closed orbits
so one often has to restrict to suitable representations of $A$ for
which it _is_ possible to construct an orbit-space. But first, let us
give a motivating example to illustrate the fact that many interesting
classification problems can be translated into the setting of this
non-commutative algebraic geometry. Let $X$ be a smooth projective
curve of genus $g$ (that is, a Riemann surface with $g$ holes). A
classical object of study is $M = M_X^{ss}(0,n)$ the _moduli space
of semi-stable vectorbundles on $X$ of rank $n$ and degree $0$_. This
space has an open subset (corresponding to the _stable_ vectorbundles)
which classify the isomorphism classes of unitary simple representations
$\pi_1(X) = \frac{\langle x_1,\ldots,x_g,y_1,\ldots,y_g
\rangle}{([x_1,y_1] \ldots [x_g,y_g])} \rightarrow U_n(\mathbb{C})$ of the
fundamental group of $X$. Let $Y$ be an affine open subset of the
projective curve $X$, then we have the formally smooth algebra $A =
\begin{bmatrix} \mathbb{C} & 0 \\ \mathbb{C}[Y] & \mathbb{C}[Y] \end{bmatrix}$ As $A$ has two
orthogonal idempotents, its representation varieties decompose into
connected components according to dimension vectors $\mathbf{rep}_m~A
= \bigsqcup_{p+q=m} \mathbf{rep}_{(p,q)}~A$ all of which are smooth
varieties. As mentioned before it is not possible to construct a
variety classifying the orbits in one of these components, but there are
two methods to approximate the orbit space. The first one is the
_algebraic quotient variety_ of which the coordinate ring is the ring of
invariant functions. In this case one merely recovers for this quotient
$\mathbf{rep}_{(p,q)}~A // GL_{p+q} = S^q(Y)$ the symmetric product
of $Y$. A better approximation is the _moduli space of semi-stable
representations_ which is an algebraic quotient of the open subset of
all representations having no subrepresentation of dimension vector
$(u,v)$ such that $-uq+vp < 0$ (that is, cover this open set by
$GL_{p+q}$ stable affine opens and construct for each the algebraic
quotient and glue them together). Denote this moduli space by
$M_{(p,q)}(A,\theta)$. It is an unpublished result of Aidan Schofield
that the moduli spaces of semi-stable vectorbundles are birational
equivalent to specific ones of these moduli spaces
$M_X^{ss}(0,n)~\sim^{bir}~M_{(n,gn)}(A,\theta)$ Rather than studying
the moduli spaces of semi-stable vectorbundles $M^{ss}_X(0,n)$ on the
curve $X$ one at a time for each rank $n$, non-commutative algebraic
geometry allows us (via the translation to the formally smooth algebra
$A$) to obtain common features on all these moduli spaces and hence to
study $\bigsqcup_n~M^{ss}_X(0,n)$ the moduli space of all
semi-stable bundles on $X$ of degree zero (but of varying ranks).
There exists a procedure to associate to any formally smooth algebra $A$
a quiver $Q_A$ (playing roughly the role of the tangent space to the
manifold determined by $A$). If we do this for the algebra described
above we find the quiver $\xymatrix{\vtx{} \ar[rr] & & \vtx{}
\ar@(ur,dr)}$ and hence the representation theory of this quiver plays
an important role in studying the geometric properties of the moduli
spaces $M^{ss}_X(0,n)$, for instance it allows to determine the smooth
loci of these varieties. Move on the the [next part.
[1]: http://www.neverendingbooks.org/index.php/quiver-representations.html
OK! I asked to get side-tracked by comments so now that there is one I’d better deal with it at once. So, is there any relation between the non-commutative (algebraic) geometry based on formally smooth algebras and the non-commutative _differential_ geometry advocated by Alain Connes?
Short answers to this question might be (a) None whatsoever! (b) Morally they are the same! and (c) Their objectives are quite different!
As this only adds to the confusion, let me try to explain each point separately after issuing a _disclaimer_ that I am by no means an expert in Connes’ NOG neither in $C^* $-algebras. All I know is based on sitting in some lectures by Alain Connes, trying at several times to make sense of his terribly written book and indeed by reading the Landi notes in utter desperation.
(a) _None whatsoever!_ : Connes’ approach via spectral triples is modelled such that one gets (suitable) ordinary (that is, commutative) manifolds into this framework. The obvious algebraic counterpart for this would be a statement to the effect that the affine coordinate ring $\mathbb{C}[X] $ of a (suitable) smooth affine variety X would be formally smooth. Now you’re in for a first shock : the only affine smooth varieties for which this holds are either _points_ or _curves_! Not much of a geometry huh? In fact, that is the reason why I prefer to call formally smooth algebras just _qurves_ …
(b) _Morally they are the same_ : If you ever want to get some differential geometry done, you’d better have a connection on the tangent bundle! Now, Alain Connes extended the notion of a connection to the non-commutative world (see for example _the_ book) and if you take the algebraic equivalent of it and ask for which algebras possess such a connection, you get _precisely_ the formally smooth algebras (see section 8 of the Cuntz-Quillen paper “Algebra extensions and nonsingularity” Journal AMS Vol 8 (1991). Besides there is a class of $C^* $-algebras which are formally smooth algebras : the AF-algebras which also feature prominently in the Landi notes (although they are virtually never affine, that is, finitely generated as an algebra).
(c) _Their objectives are quite different!_ : Connes’ formalism aims to define a length function on a non-commutative manifold associated to a $C^* $-algebra. Non-commutative geometry based on formally smooth algebras has no interest in defining some sort of space associated to the algebra. The major importance of formally smooth algebras (as advocated by Maxim Kontsevich is that such an algebra A can be seen as a _machine_ producing an infinite family of ordinary commutative manifolds via its _representation varieties_ $\mathbf{rep}_n~A $ which are manifolds equipped with a $GL_n $-action. Non-commutative functions and diifferential forms defined at the level of the formally smooth algebra A do determine similar $GL_n $-invariant features on _all_ of these representation varieties at once.