In this

series of posts I’ll try to make at least part of the recent

[Kontsevich-Soibelman paper](http://www.arxiv.org/abs/math.RA/0606241) a

bit more accessible to algebraists. In non-geometry, the algebras

corresponding to *smooth affine varieties* I’ll call **qurves** (note

that they are called **quasi-free algebras** by Cuntz & Quillen and

**formally smooth** by Kontsevich). By definition, a qurve in an affine

$\mathbb{C} $-algebra A having the lifting property for algebra

maps through nilpotent ideals (extending Grothendieck’s characterization

of smooth affine algebras in the commutative case). Examples of qurves

are : finite dimensional semi-simple algebras (for example, group

algebras $\mathbb{C} G $ of finite groups), coordinate rings of

smooth affine curves or a noncommutative mixture of both, skew-group

algebras $\mathbb{C}[X] \ast G $ whenever G is a finite group of

automorphisms of the affine curve X. These are Noetherian examples but

in general a qurve is quite far from being Noetherian. More typical

examples of qurves are : free algebras $\mathbb{C} \langle

x_1,\ldots,x_k \rangle $ and path algebras of finite quivers

$~\mathbb{C} Q $. Recall that a finite quiver Q s just a

directed graph and its path algebra is the vectorspace spanned by all

directed paths in Q with multiplication induced by concatenation of

paths. Out of these building blocks one readily constructs more

involved qurves via universal algebra operations such as (amalgamated)

free products, universal localizations etc. In this way, the

groupalgebra of the modular group $SL_2(\mathbb{Z}) $ (as well

as that of a congruence subgroup) is a qurve and one can mix groups with

finite groupactions on curves to get qurves like $ (\mathbb{C}[X]

\ast G) \ast_{\mathbb{C} H} \mathbb{C} M $ whenever H is a common

subgroup of the finite groups G and M. So we have a huge class of

qurve-examples obtained from mixing finite and arithmetic groups with

curves and quivers. Qurves can we used as *machines* generating

interesting $A_{\infty} $-categories. Let us start by recalling

some facts about finite closed subschemes of an affine smooth variety Y

in the commutative case. Let **fdcom** be the category of all finite

dimensional commutative $\mathbb{C} $-algebras with morphisms

being onto algebra morphisms, then the study of finite closed subschemes

of Y is essentially the study of the covariant functor **fdcom** –>

**sets** assigning to a f.d. commutative algebra S the set of all onto

algebra maps from $\mathbb{C}[Y] $ to S. S being a f.d.

commutative semilocal algebra is the direct sum of local factors $S

\simeq S_1 \oplus \ldots \oplus S_k $ where each factor has a

unique maximal ideal (a unique point in Y). Hence, our study reduces to

f.d. commutative images with support in a fixed point p of Y. But all

such quotients are also quotients of the completion of the local ring of

Y at p which (because Y is a smooth variety, say of dimension n) is

isomorphic to formal power series

$~\mathbb{C}[[x_1,\ldots,x_n]] $. So the local question, at any

point p of Y, reduces to finding all settings

$\mathbb{C}[[x_1,\ldots,x_n]] \twoheadrightarrow S

\twoheadrightarrow \mathbb{C} $ Now, we are going to do something

strange (at least to an algebraist), we’re going to take duals and

translate the above sequence into a coalgebra statement. Clearly, the

dual $S^{\ast} $ of any finite dimensional commutative algebra

is a finite dimensional cocommutative coalgebra. In particular

$\mathbb{C}^{\ast} \simeq \mathbb{C} $ where the

comultiplication makes 1 into a grouplike element, that is

$\Delta(1) = 1 \otimes 1 $. As long as the (co)algebra is

finite dimensional this duality works as expected : onto maps correspond

to inclusions, an ideal corresponds to a sub-coalgebra a sub-algebra

corresponds to a co-ideal, so in particular a local commutative algebra

corresponds to an pointed irreducible cocommutative coalgebra (a

coalgebra is said to be irreducible if any two non-zero subcoalgebras

have non-zero intersection, it is called simple if it has no non-zero

proper subcoalgebras and is called pointed if all its simple

subcoalgebras are one-dimensional. But what about infinite dimensional

algebras such as formal power series? Well, here the trick is not to

take all dual functions but only those linear functions whose kernel

contains a cofinite ideal (which brings us back to the good finite

dimensional setting). If one takes only those good linear functionals,

the ‘fancy’-dual $A^o $of an algebra A is indeed a coalgebra. On

the other hand, the full-dual of a coalgebra is always an algebra. So,

between commutative algebras and cocommutative coalgebras we have a

duality by associating to an algebra its fancy-dual and to a coalgebra

its full-dual (all this is explained in full detail in chapter VI of

Moss Sweedler’s book ‘Hopf algebras’). So, we can dualize the above pair

of onto maps to get coalgebra inclusions $\mathbb{C} \subset

S^{\ast} \subset U(\mathfrak{a}) $ where the rightmost coalgebra is

the coalgebra structure on the enveloping algebra of the Abelian Lie

algebra of dimension n (in which all Lie-elements are primitive, that is

$\Delta(x) = x \otimes 1 + 1 \otimes x $ and indeed we have that

$U(\mathfrak{a})^{\ast} \simeq \mathbb{C}[[x_1,\ldots,x_n]] $.

We have translated our local problem to finding all f.d. subcoalgebras

(containing the unique simple) of the enveloping algebra. But what is

the point of this translation? Well, we are not interested in the local

problem, but in the global problem, so we somehow have to **sum over all

points**. Now, on the algebra level that is a problem because the sum of

all local power series rings over all points is no longer an algebra,

whereas the direct sum of all pointed irreducible coalgebras $~B_Y

= \oplus_{p \in Y} U(\mathfrak{a}_p) $ is again a coalgebra! That

is, we have found a huge coalgebra (which we call the coalgebra of

‘distributions’ on Y) such that for every f.d. commutative algebra S we

have $Hom_{comm alg}(\mathbb{C}[Y],S) \simeq Hom_{cocomm

coalg}(S^{\ast},B_Y) $ Can we get Y back from this coalgebra of

districutions? Well, in a way, the points of Y correspond to the

group-like elements, and if g is the group-like corresponding to a point

p, we can recover the tangent-space at p back as the g-primitive

elements of the coalgebra of distributions, that is the elements such

that $\Delta(x) = x \otimes g + g \otimes x $. Observe that in

this commutative case, there are no **skew-primitives**, that is

elements such that $\Delta(x) = x \otimes g + h \otimes x $ for

different group-likes g and h. This is the coalgebra translation of the

fact that a f.d. semilocal commutative algebra is the direct sum of

local components. This is something that will definitely change if we

try to extend the above to the case of qurves (to be continued).