If you ever sat through a lecture by Alain Connes you will know about his insistence on the ‘canonical dynamic nature of noncommutative manifolds’. If you haven’t, he did write a blog post Heart bit 1 about it.

I’ll try to explain here that there is a definite “supplément d’âme” obtained in the transition from classical (commutative) spaces to the noncommutative ones. The main new feature is that “noncommutative spaces generate their own time” and moreover can undergo thermodynamical operations such as cooling, distillation etc…

Here a section from his paper A view of mathematics :

Indeed even at the coarsest level of understanding of a space provided by measure theory, which in essence only cares about the “quantity of points” in a space, one finds unexpected completely new features in the noncommutative case. While it had been long known by operator algebraists that the theory of von-Neumann algebras represents a far reaching extension of measure theory, the main surprise which occurred at the beginning of the seventies is that such an algebra M inherits from its noncommutativity a god-given time evolution: where is the quotient of the group of automorphisms of M by the normal subgroup of inner automorphisms. This led in my thesis to the reduction from type III to type II and their automorphisms and eventually to the classification of injective factors.

Even a commutative manifold has a kind of dynamics associated to it. Take a suitable vectorfield, consider the flow determined by it and there’s your ‘dynamics’, or a one-parameter group of automorphisms on the functions. Further, other classes of noncommutative algebras have similar features. For example, Cuntz and Quillen showed that also formally smooth algebras (the noncommutative manifolds in the algebraic world) have natural Yang-Mills flows associated to them, giving a one-parameter subgroup of automorphisms.

Let us try to keep far from mysticism and let us agree that by ‘time’ (let alone ‘god given time’) we mean a one-parameter subgroup of algebra automorphisms of the noncommutative algebra. In nice cases, such as some von-Neumann algebras this canonical subgroup is canonical in the sense that it is unique upto inner automorphisms.

In the special case of the Bost-Connes algebra these automorphisms are given by and .

This one-parameter subgroup is crucial in the definition of the so called KMS-states (for Kubo-Martin and Schwinger) which is our next goal.

Yesterday, Ed Segal gave a talk at the Arts. His title “Superpotential algebras from 3-fold singularities” didnt look too promising to me. And sure enough it was all there again : stringtheory, D-branes, Calabi-Yaus, superpotentials, all the pseudo-physics babble that spreads virally among the youngest generation of algebraists and geometers.

Fortunately, his talk did contain a general ringtheoretic gem. After a bit of polishing up this gem, contained in his paper The A-infinity Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus, can be stated as follows.

Let be a -algebra and let be a finite dimensional semi-simple representation with distinct simple components. Let be the kernel of the algebra epimorphism to the semi-simple algebra . Then, the -adic completion of is Morita-equivalent to the completion of a quiver-algebra with relations. The nice thing is that both the quiver and relations come in a canonical way from the -structure on the Ext-algebra . More precisely, there is an isomorphism

where the homotopy Maurer-Cartan map comes from the structure maps

and hence the defining relations of the completion are given by the image of the dual of this map.

For ages, Ive known this result in the trivial case of formally smooth algebras (where and hence there are no relations to divide out) and where it is a consequence of a special case of the Cuntz-Quillen “tubular neighborhood” result. Completions of formally smooth algebras at semi-simples are Morita equivalent to completions of path algebras. This fact motivated all the local-quiver technology that was developed here in Antwerp over the last decade (see my book if you want to know the details).

Also for 3-dimensional Calabi-Yau algebras it states that the completions at semi-simples are Morita equivalent to completions of quotients of path algebras by the relations coming from a superpotential (aka a necklace) by taking partial noncommutative derivatives. Here the essential ingredient is that in this case.

Last time we introduced the tangent quiver of an affine algebra A to be a quiver on the isoclasses of simple finite dimensional representations. When is the coordinate ring of an affine variety, these vertices are just the points of the variety and this set has the extra structure of being endowed with the Zariski topology. For a general, possibly noncommutative algebra, we would like to equip the vertices of also with a topology.

In the commutative case, the Zariski topology has as its closed sets the common zeroes of a set of polynomials on , so we need to generalize the notion of ‘functions’ the the noncommutative world. The NC-mantra states that we should view the algebra A as the ring of functions on a (usually virtual) noncommutative space. And, face it, for a commutative variety the algebra does indeed do the job. Still, this is a red herring.

Let’s consider the easiest noncommutative case, that of the group algebra of a finite group . In this case, the vertices of the tangent quiver are the irreducible representations of and no sane person would consider the full group algebra to be the algebra of functions on this set. However, we do have a good alternative in this case : characters which allow us to separate the irreducibles and are a lot more manageable than the full group algebra. For example, if is the monster group then the group algebra has dimension approx whereas there are just 194 characters to consider…

But, can we extend characters to arbitrary noncommutative algebras? and, more important, are there enough of these to separate the simple representations? The first question is easy enough to answer, after all characters are just traces so we can define for every element and any finite dimensional simple A-representation the character

where is the matrix describing the action of a on S. But, you might say, characters are then just linear functionals on the algebra A so it is natural to view A as the function algebra, right? Wrong! Traces have the nice property that and so they vanish on all commutators of A, so characters only carry information of the quotient space

where is the vectorspace spanned by all commutators (and not the ideal…). If one is too focussed on commutative geometry one misses this essential simplification as clearly for being a commutative algebra,

and therefore in this case

Ok, but are there enough characters (that is, linear functionals on , that is elements of the dual space ) to separate the simple representations? And, why do I (ab)use Lie-algebra notation to denote the vectorspace ???

Last time we have seen that the coalgebra of distributions of an affine smooth variety is the direct sum (over all points) of the dual to the etale local algebras which are all of the form where is the dimension of the variety. Generalizing this to non-commutative manifolds, the first questions are : “What is the analogon of the power-series algebra?” and do all ‘points’ of our non-commutative manifold do have such local algebras? Surely, we no longer expect the variables to commute, so a non-commutative version of the power series algebra should be , the ring of formal power series in non-commuting variables. However, there is still another way to add non-commutativity and that is to go from an algebra to matrices over the algebra. So, in all we would expect to be our local algebras at points of our non-commutative manifold to be isomorphic to As to the second question : _qurves (that is, the coordinate rings of non-commutative manifolds) do have such algebras as local rings provided we take as the ‘points’ of the non-commutative variety the set of all simple finite dimensional representations of the qurve. This is a consequence of the tubular neighborhood theorem due to Cuntz and Quillen. In more details : If A is a qurve, then a simple -dimensional representation corresponds to an epimorphism and if we take , then is an -bimodule and the -adic completion of A is isomorphic to the completed tensor-algebra In contrast with the commutative case however where the dimension remains constant over all points, here the numbers n and d can change from simple to simple. For n this is clear as it gives the dimension of the simple representation, but also d changes (it is the local dimension of the variety classifying simple representations of the same dimension). Here an easy example : Consider the skew group algebra with the action given by sending . Then A is a qurve and its center is with . Over any point there is a unique simple 2-dimensional representation of A giving the local algebra . If the situation is more complicated as the local structure of A is given by the algebra So, over this point there are precisely 2 one-dimensional simple representations corresponding to the maximal ideals and both ideals are idempotent, that is whence the corresponding bimodule so the local algebra in either of these two points is just . Ok, so the comleted local algebra at each point is of the form , but what is the corresponding dual coalgebra. Well, is the algebra dual to the _cofree coalgebra on . As a vectorspace this is the tensor-algebra with the coalgebra structure induced by the bialgebra structure defined by taking all varaibales to be primitives, that is . That is, the coproduct on a monomial gives all different expressions such that . For example, . On the other hand, the dual coalgebra of is the matrix coalgebra which is the -dimensional vectorspace with comultiplication The coalgebra corresponding to the local algebra is then the tensor-coalgebra of the matrix coalgebra and the cofree coalgebra. Having obtained the coalgebra at each point (=simple representation) of our noncommutative manifold one might think that the _coalgebra of non-commutative distributions should be the direct sum of all this coalgebras, summed over all points, as in the commutative case. But then we would forget about a major difference between the commutative and the non-commutative world : distinct simples can have non-trivial extensions! The mental picture one might have about simples having non-trivial extensions is that these points lie ‘infinitesimally close’ together. In the example above, the two one-dimensional simples have non-trivial extensions so they should be thought of as a cluster of two infinitesimally close points corresponding to the point (that is, this commutative points splits into two non-commutative points). Btw. this is the reason why non-commutative algebras can be used to resolve commutative singularities (excessive tangents can be split over several non-commutative points). While this is still pretty harmless when the algebra is finite over its center (as in the above example where only the two one-dimensionals have extensions), the situation becomes weird over general qurves as ‘usually’ distinct simples have non-trivial extensions. For example, for the free algebra this is true for all simples… So, if we want to continue using this image of points lying closely together this immediately means that non-commutative ‘affine’ manifolds behave like compact ones (in fact, it turns out to be pretty difficult to ‘glue’ together qurves into ‘bigger’ non-commutative manifolds, apart from the quiver examples of this old paper). So, how to bring this new information into our coalgebra of distributions? Well, let’s repeat the previous argument not with just one point but with a set of finitely many points. Then we have a _semi-simple algebra quotient and taking again and , then is again an S-bimodule. Now, any S-bimodule can be encoded into a quiver Q on k points, the number of arrows from vertex i to vertex j being the number of components in M of the form . Again, it follows from the tubular neighborhood theorem that the -adic completion of A is isomorphic to the completion of an algebra Morita equivalent to the _path algebra (being the tensor algebra ). As all the local algebras of the points are quotients of this quiver-like completion, on the coalgebra level our local coalgebras will be sub coalgebras of the coalgebra which is co-Morita equivalent (and believe it or not but coalgebraists have a name for this : _Takeuchi equivalence) to the quiver coalgebra which is the vectorspace of the path algebra with multiplication induced by making all arrows from i to j skew-primitives, that is, where the are group-likes corresponding to the vertices. If all of ths is a bit too much co to take in at once, I suggest the paper by Bill Chin A brief introduction to coalgebra representation theory. The _coalgebra of noncommutative distributions we are after at is now the union of all these Takeuchi-equivalent quiver coalgebras. In easy examples such as the -example this coalgebra is still pretty small (the sum of the local coalgebras corresponding to the local algebras summed over all points summed with the quiver coalgebra of the quiver In general though this is a huge object and we would like to have a recipe to construct it from a manageable blue-print and that is what we will do next time.

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 -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 of finite groups), coordinate rings of smooth affine curves or a noncommutative mixture of both, skew-group algebras 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 and path algebras of finite quivers . 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 (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 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 -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 -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 to S. S being a f.d. commutative semilocal algebra is the direct sum of local factors 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 . So the local question, at any point p of Y, reduces to finding all settings 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 of any finite dimensional commutative algebra is a finite dimensional cocommutative coalgebra. In particular where the comultiplication makes 1 into a grouplike element, that is . 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 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 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 and indeed we have that . 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 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 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 . Observe that in this commutative case, there are no skew-primitives, that is elements such that 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).