Cluster-Algebra

In mathematics cluster algebras are used inter alia in representation theory, low -dimensional topology and Higher Teichmüller theory. Cluster algebras are subalgebras of, given by producers, which are summarized in n-element " clusters " with given by skew-symmetric exchange matrices transition rules (called mutations) between clusters.

They were introduced in 2002 by Andrei Zelevinsky and Sergey Fomin.

Definition

A cluster is a pair of

• An n- tuple algebraically independent variables,
• A skew-symmetric, integer matrix, the exchange matrix.

For the mutation is defined by using

Is also a cluster, are involutions.

A cluster algebra arises from a cluster by iterated application of all possible mutations. The cluster algebra is called of finite type if there are only finitely many clusters.

Examples

A1

For the skew-symmetric matrix must be calculated to

Because this is a cluster algebra of finite type, it corresponds to the Cartan matrix.

A2

Be and. We calculate

This cluster algebra is therefore of finite type, it corresponds to the Cartan matrix.

For and obtained cluster algebras infinite type.

Cluster algebras topological origin

A triangulated surface defined assigned to a cluster algebra as follows:

• The variables are the edges of the triangulation,
• If the i-th and j-th edge follow each other within a triangle in the direction,
• If the j-th and i-th edge follow each other within a triangle in the direction,
• Else

Generally, one can cluster algebras also in possibly degenerate triangles decomposed surfaces associate (see the work of Fomin - Shapiro - Thurston ), the resulting cluster algebras are called cluster algebras topological origin.

The mutations in this case are given by flips of the edges of the triangulation, ie an edge is considered, the plane defined by the two triangles are adjacent rectangle and then replaced by the edge of the other diagonal of this quadrilateral.

Cluster algebras of finite type

Fomin and Zelevinsky proved that there is a bijection between cluster algebras of finite type and Cartan matrices of finite type. Cluster algebras of finite type are therefore classified by Dynkin diagrams. The Cartan matrices can be calculated from the exchange matrices.

Felikson, Shapiro and Tumarkin proved that cluster algebras of finite type cluster algebras are either topological origin or equivalent to one of 11 exceptional algebras.