In what way is a formally smooth algebra a machine producing families of manifolds? Consider the special case of the path algebra $\C Q$ of a quiver and recall that an $n$-dimensional representation is an algebra map $\C Q \rightarrow^{\phi} Mn(\C)$ or, equivalently, an $n$-dimensional left $\C Q$-module $\C^n{\phi}$ with the action determined by the rule $a.v = \phi(a) v~\forall v \in \C^n{\phi}, \forall a \in \C Q$ If the $ei~1 \leq i \leq k$ are the idempotents in $\C Q$ corresponding to the vertices (see this [post][1]) then we get a direct sum decomposition $\C^n{\phi} = \phi(e1)\C^n{\phi} \oplus \hdots \oplus \phi(ek)\C^n{\phi}$ and so every $n$-dimensional representation does determine a _dimension vector $\alpha = (a1,\hdots,ak)~\text{with}~ai = dim{\C} Vi = dim{\C} \phi(ei)\C^n{\phi}$ with $ | \alpha | = \sumi ai = n$. Further, for every arrow we have (because $ej.a.ei = a$ that $\phi(a)$ defines a linear map $\phi(a)~:~Vi \rightarrow Vj$ (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 $Vi$ 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 $\wis{rep}{\alpha}~Q = \oplusa~M{aj \times ai}(\C)$ and base-change in the vertex-spaces does determine the action of the base-change group $GL(\alpha) = GL{a1} \times \hdots \times GL{ak}$ on this space. Finally, as all this started out with fixing a bases in $\C^n{\phi}$ we get the affine variety of all $n$-dimensional representations by bringing in the base-change $GLn$-action (by conjugation) and have $\wis{rep}n~\C Q = \bigsqcup{| \alpha | = n} GLn \times^{GL(\alpha)} \wis{rep}{\alpha}~Q$ 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 $GLn$-orbits in the fiber-bundle $GLn \times^{GL(\alpha)} \wis{rep}{\alpha}~Q$ and $GL(\alpha)$-orbits in the affine space $\wis{rep}{\alpha}~Q$. In our main example
an $n$-dimensional representation determines vertex-spaces $V = \phi(e) \C^n{\phi}$ and $W = \phi(f) \C^n{\phi}$ of dimensions $p,q~\text{with}~p+q = n$. The arrows determine linear maps between these spaces and if we fix a set of bases in these two vertex-spaces, we can represent these maps by matrices
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 $GL(\alpha) = GLp \times GLq$ is the diagonal subgroup of $GLn$
$GL(\alpha) = \begin{bmatrix} GLp & 0 \ 0 & GLq \end{bmatrix}$ and acts on the representation space $\wis{rep}{\alpha}~Q = M{q \times p}(\C) \oplus M{p \times q}(\C) \oplus Mq(\C) \oplus Mq(\C)$ (embedded as block-matrices in $Mn(\C)^{\oplus 4}$ as above) by simultaneous conjugation. More generally, if $A$ is a formally smooth algebra, then all its representation varieties $\wis{rep}n~A$ are affine smooth varieties equipped with a $GLn$-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 $\phi~:~\C Q \rightarrow B/I$ 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 $\psi(ej)\pi^{-1}(\phi(a))\psi(ei)$. 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 $\wis{rep}_n~A$ 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

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1 comment
Posted in geometry
Written on Mon, 06 September 2004 at 12:24 pm
Tags: arxiv, geometry, Grothendieck, mac, non-commutative, representations
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