Applications of cluster algebras

Question

Why are so many algebraists nowadays interested in cluster algebras?

(This is a rewording of one half of the closed question Cluster algebras and teichmuller theory.)

Answer 1

One reason is that cluster algebras have motivated many recent developments in the representation theory of associative algebras. There is a lot one can say about this, so I will try to just give an overview of some of the key ideas, and suggest further reading. I recommend Keller's survey article (https://arxiv.org/abs/0807.1960) as a good source for a more detailed discussion.

A simple case of a cluster algebra is the cluster algebra without frozen variables defined by a skew-symmetric matrix. A skew-symmetric matrix is essentially the same data (modulo some technicalities) as a quiver (directed graph) without loops or $2$-cycles, e.g.

$$\begin{pmatrix}0&-2\\2&0\end{pmatrix}\ \longleftrightarrow\ 1\Rightarrow 2$$

where the '$\Rightarrow$' represents a pair of arrows. The matrix here is the skew-symmetrisation of the adjacency matrix of the quiver, the assumptions on loops and $2$-cycles ensuring that there is no cancellation when computing this matrix, so that the process is invertible. (For simplicity, to avoid algebras breaking up into direct products, I consider only the pairs in the above correspondence such that the quiver is connected; this is most important in the classification statements below.) Such quivers, when they have no oriented cycles, also define associative algebras; they correspond to the finite dimensional basic hereditary algebras over a field $k$ via $Q\leftrightarrow kQ$ (I take $k$ to be algebraically closed, just in case). The algebra $kQ$ has basis given by the paths of $Q$ (including the length $0$ paths at each vertex) with multiplication given by concatenation of paths when the result is a path and $0$ otherwise, extended via linearity.

A classical result of Gabriel says that the algebra $kQ$ has finitely many isomorphism classes of indecomposable modules if and only if $Q$ is an orientation of one of the simply laced Dynkin graphs, i.e. those of type $\mathsf{A}_n$, $\mathsf{D}_n$ or $\mathsf{E}_{6,7,8}$. Suggestively, Fomin–Zelevinsky prove in their second paper on cluster algebras that the cluster algebras (without frozen variables, defined by a skew-symmetric matrix) with finitely many cluster variables are precisely those from matrices corresponding to these quivers.

It turns out that, whenever $Q$ is an acyclic quiver, there is in fact a bijection between most of the cluster variables (precisely, those not appearing in the initial seed) of the cluster algebra $\mathcal{A}_Q$ given by $Q$, and the indecomposable representations of $kQ$.

To make this connection stronger, one can introduce the 'cluster category' $\mathcal{C}_Q$, defined by Buan, Marsh, Reineke, Reiten and Todorov (https://arxiv.org/abs/math/0402054), which is a kind of extension of the category of $kQ$-modules by adding in some extra indecomposable objects corresponding to the initial cluster variables of $\mathcal{A}_Q$, so that each cluster variable $x$ of $\mathcal{A}_Q$ now corresponds to an indecomposable object $M_x$ of $\mathcal{C}_Q$.

Various properties of the cluster variables may now be translated into properties of these objects. For example, two cluster variables $x$ and $y$ are compatible (appear in the same cluster) if and only if the corresponding objects have the homological property $\operatorname{Ext}^1_{\mathcal{C}_Q}(M_x,M_y)=0$. The seeds of $\mathcal{A}_Q$ correspond to basic cluster-tilting objects of $\mathcal{C}_Q$, which are direct sums $T$ of some of the $M_x$s such that $\operatorname{Ext}^1_{\mathcal{C}_Q}(T,M_x)=0$ if and only if $M_x$ appears as a summand of $T$.

The quiver of the seed corresponding to such an object $T$ is the ordinary quiver of the endomorphism algebra $\operatorname{End}_{\mathcal{C}_Q}(T)$, i.e. the unique (up to isomorphism) quiver $Q_T$ such that $\operatorname{End}_{\mathcal{C}_Q}(T)\cong kQ_T/I$ for some ideal generated by linear combinations of paths of length at least $2$. The category $\mathcal{C}_Q$ is a '$2$-Calabi–Yau triangulated category', so its cluster-tilting objects have a mutation theory by work of Iyama–Yoshino (https://arxiv.org/abs/math/0607736) which turns out to correspond to mutations of seeds.

There is considerably more work in this direction, including cluster categories for cluster algebras corresponding to non-acyclic quivers by Amiot (https://arxiv.org/abs/0805.1035), and for cluster algebras with frozen variables by, e.g. Geiß–Leclerc–Schröer (https://arxiv.org/abs/math/0609138), Jensen–King–Su (https://arxiv.org/abs/1309.7301), Demonet–Iyama (https://arxiv.org/abs/1503.02362) and myself (https://arxiv.org/abs/1510.06224, https://arxiv.org/abs/1702.05352). One can also develop some theory for skew-symmetrizable matrices, see e.g. Demonet (https://arxiv.org/abs/0909.1633), Labardini-Fragoso–Zelevinsky (https://arxiv.org/abs/1306.3495) and Geiß–Leclerc–Schröer again (https://arxiv.org/abs/1410.1403).

While construction of these kinds of representation-theoretic models is useful for understanding cluster algebras, it also suggests lots of interesting representation theory that may not otherwise have been considered. This can make a precise connection difficult to pin down: I am not really aware of cluster algebras being used directly to solve open representation-theoretic problems, but they have suggested lots of new lines of representation-theoretic inquiry (e.g. the second reference to Geiß–Leclerc–Schröer above, which I think is not really about cluster algebras at all, a priori).

One example of this is Adachi–Iyama–Reiten's $\tau$-tilting theory (https://arxiv.org/abs/1210.1036). I am far from an expert on this theory, but my understanding is that this is really a purely representation-theoretic theory that could have been developed entirely without cluster algebras (and, I am told, was almost discovered by Auslander and co-authors in the 80s!) but in the end it was, at least in part, thinking about cluster categories that provided the inspiration.

I note that one slightly confusing part of this connection is that one quite rarely studies the cluster algebra itself as an algebra! There are some exceptions to this, such as work of Lampe (https://arxiv.org/abs/1210.1502).