Cluster algebra

Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky (2002, 2003, 2007).

A cluster of rank n consists of a set of n elements {x, y, ...} of F, usually assumed to be an algebraically independent set of generators of a field extension F. A seed consists of a cluster {x, y, ...} of F, together with an exchange matrix B with integer entries bx,y indexed by pairs of elements x, y of the cluster.

More generally the matrix might be skew-symmetrizable, meaning there are positive integers dx associated with the elements of the cluster such that dxbx,y = –dyby,x for all x and y.

It is common to picture a seed as a quiver whose vertices are the generating set, by drawing bx,y arrows from x to y if this number is positive.

Finally replace y by a new generator w, where where the products run through the elements t in the cluster of the seed such that bt,y is positive or negative respectively.

The cluster algebra also comes with the extra structure of the seeds of this graph.

Fomin & Zelevinsky (2003) showed that the cluster algebras of finite type can be classified in terms of the Dynkin diagrams of finite-dimensional simple Lie algebras.

Suppose that we start with the cluster {x1, x2} and take the exchange matrix with b12 = –b21 = 1.

Then mutation gives a sequence of variables x1, x2, x3, x4,... such that the clusters are given by adjacent pairs {xn, xn+1}.

There are similar examples with b12 = 1, –b21 = 2 or 3, where the analogous sequence of cluster variables repeats with period 6 or 8.

Simple examples are given by the algebras of homogeneous functions on the Grassmannians.

In that case, the Plücker coordinates provide all the distinguished elements and the clusters can be completely described using triangulations of a regular polygon with n vertices.

More precisely, clusters are in one-to-one correspondence with triangulations and the distinguished elements are in one-to-one correspondence with diagonals (line segments joining two vertices of the polygon).

Suppose S is a compact connected oriented Riemann surface and M is a non-empty finite set of points in S that contains at least one point from each boundary component of S (the boundary of S is not assumed to be either empty or non-empty).

The pair (S, M) is often referred to as a bordered surface with marked points.

are in the Weyl group) there are cluster coordinate charts depending on reduced word decompositions of

These are called factorization parameters and their structure is encoded in a wiring diagram.