Which cluster algebras have been categorified?

Question

In "Tilting Theory and Cluster Combinatorics" Buan, Marsh, Reineke, Reiten, and Todorov constructed cluster categories for mutation finite cluster algebras (without coefficients), and Amiot gives a construction of a cluster category given a quiver with potential whose Jacobian algebra is finite dimensional (in particular, this gives a cluster category for cluster algebras coming from unpunctured surfaces with nonempty boundary).

My question is simply in what other instances have cluster categories been constructed? In particular, what about cluster algebras with coefficients? What about other surface cluster algebras? What about tame cluster algebras?

Answer 1

There are several different questions in here and what follows are only partial answers to some of these, mostly consisting of pointers to pieces of the literature.

• "In what other instances have cluster categories been constructed?"

For surveys on cluster categories, I would recommend:

Cluster algebras, quiver representations and triangulated categories, Bernhard Keller, arXiv:0807.1960

Categorification of acyclic cluster algebras: an introduction, Bernhard Keller, arXiv:0801.3103

Cluster categories, Idun Reiten, arXiv:1012.4949 (given as an ICM 2010 lecture)

• Cluster algebras with coefficients

These don't fall inside the usual cluster category setting, as the coefficients (frozen variables) are expected to correspond to projective objects and typically cluster category theory works at the stable level, where these are not present. (I am not an expert on 2-Calabi-Yau categories, etc., so if someone else wishes to tidy up this claim a little, that would be helpful.) So one suggestion has been that one ought to work in a Frobenius category whose stable category would give the usual cluster category. This corresponds to the theorems that say that for each cluster algebra type, you can have different cluster algebras with different numbers of coefficients present but they all have the same cluster combinatorics.

Work on getting such categories is in its early stages but being actively pursued by a number of researchers. One very notable success to date is the work of Geiss-Leclerc-Schroer on cluster algebra structures on coordinate rings associated to partial flag varieties:

Partial flag varieties and preprojective algebras, Christof Geiss, Bernard Leclerc, Jan Schröer, arXiv:math/0609138

Preprojective algebras and cluster algebras, Christof Geiss, Bernard Leclerc, Jan Schröer, arXiv:0804.3168 (a survey article covering the previously-listed paper and some earlier ones)

Kac-Moody groups and cluster algebras, Christof Geiss, Bernard Leclerc, Jan Schröer, arXiv:1001.3545 (generalization of the above)

Cluster algebras and representation theory, Bernard Leclerc, arXiv:1009.4552 (another more recent survey, from the 2010 ICM)

• Surface and tame cluster algebras

For things to do with more general surface cluster algebras, I suggest you start with this survey:

Geometric construction of cluster algebras and cluster categories, Karin Baur, arXiv:0804.4065

I don't know enough about tame cluster algebras to say anything about these, other than noting that for example the work of Geiss-Leclerc-Schroer mentioned above includes many cluster algebras that are not of finite type.

• Other comments

I would also add that there is work going on on quantum versions of these and infinite versions (see for example a paper by Holm and Jorgensen, arXiv:0902.4125).

Also, one can look at your question from a different angle, namely to ask which algebras occuring "in nature" admit (possibly quantum) cluster algebra structures; any good search engine will show you that this is the area I work in... Then the question becomes (a) are these categorified just as algebras and (b) are they categorified in a way that is compatible with the cluster structure? For example, Geiss-Leclerc-Schroer says "yes" to both for a large class of important examples.

Answer 2

Jan's answer includes many excellent references. I will try to give a few quick comments.

First of all, although the original Buan-Marsh-Reineke-Reiten-Todorov paper contained some results which were restricted to finite type cluster algebras, in subsequent work of them and others, notably Caldero-Keller, it has been shown that the BMRRT cluster category does indeed categorify any acyclic cluster algebra without coefficients (in particular, including those of tame type).

Although categorifications are often done in coefficient-free settings, one can also "cheat" by categorifying the cluster algebra in which the coefficients are treated as variables. For example, this allows one to treat the case of principal coefficients (in some sense, the most important case) for an acyclic cluster algebra using the BMRRT technology.

I should also point out that the categorifications in question here are what are now sometimes called additive categorifications; Hernandez-Leclerc and Nakajima have a multiplicative categorification which is more in the spirit of what people mean by categorification in other settings: they construct a category with direct sum and tensor product, and show the ring structure induced on the Grothendieck group of their categories is naturally a cluster algebra.