coalgebras and non-geometry

By lieven

In this series of posts I’ll try to make at least part of the recent Kontsevich-Soibelman paper 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 $\mathbb{C} \langle x_1,\hdots,x_k \rangle$ 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 $S \simeq S_1 \oplus \hdots \oplus S_k$ 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 $~\mathbb{C}[[x_1,\hdots,x_n]]$ . So the local question, at any point p of Y, reduces to finding all settings $\mathbb{C}[[x_1,\hdots,x_n]] \twoheadrightarrow S \twoheadrightarrow \mathbb{C}$ 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 $U(\mathfrak{a})^{\ast} \simeq \mathbb{C}[[x_1,\hdots,x_n]]$ . 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 $Hom_{comm alg}(\C[Y],S) \simeq Hom_{cocomm coalg}(S^{\ast},B_Y)$ 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).