Cluster algebra
Encyclopedia
Cluster algebras are a class of commutative ring
s introduced by . A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose union generates the algebra A and which satisfy various conditions.
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. The matrix is sometimes assumed to be skew symmetric, so that bx,y = –by,x. 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. It is common to picture a seed as a quiver
with vertices the generating set, by drawing bx,y arrows from x to y if this number is positive. When bx,y is skew symmetrizable the quiver has no loops or 2-cycles.
A mutation of a seed, depending on a choice of vertex y of the cluster, is a new seed given by a generalization of tilting
as follows. Exchange the values of bx,y and by,x for all x in the cluster. If bx,y > 0 arrows and by,z > 0 then replace bx,z by bx,yby,z + bx,z. 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 inverse of a mutation is also a mutation: in other words, if A is a mutation of B, then B is a mutation of A.
A cluster algebra is constructed from a seed as follows.
If we repeatedly mutate the seed in all possible ways, we get a finite or infinite graph of seeds, where two seeds are joined if one can be obtained by mutating the other. The underlying algebra of the cluster algebra is the algebra generated by all the clusters of all the seeds in this graph. The cluster algebra also comes with the extra structure of the seeds of this graph.
A cluster algebra is said to be of finite type if it has only a finite number of seeds. showed that the cluster algebras of finite type can be classified in terms of the Dynkin diagrams of finite dimensional simple Lie algebras.
so are given by the sequence
which repeats with period 5. So this cluster algebra has exactly 5 clusters, and in particular is of finite type. It is associated with the Dynkin diagram A2.
There are similar examples with b12 = 1, –b21 = 2 or 3, where the analogous sequence of cluster variables repeats with period 6 or 8. These are also of finite type, and are associated with the Dynkin diagrams B2 and G2. However if |b12b21| ≥ 4 then the sequence of cluster variables is not periodic and the cluster algebra is of infinite type.
There are 6 cluster variables other than the 3 initial ones x1, x2, x3 given by
.
They correspond to the 6 positive roots of the Dynkin diagram A3: more precisely the denominators are monomials in x1, x2, x3, corresponding to the expression of positive roots as the sum of simple roots.
The 3+6 cluster variables generate a cluster algebra of finite type, associated with the Dynkin diagram A3.
The 14 clusters are the vertices of the cluster graph, which is an associahedron
.
s. The Plücker coordinates provide some of the distinguished elements.
For the Grassmannian of planes in ℂn, the situation is even more simple. In that case, the Plücker coordinates provide all the distinguished elements and the clusters can be completely described using triangulation
s 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). One can distinguish between diagonals in the boundary, which belong to every cluster, and diagonals in the interior. This corresponds to a general distinction between coefficient variables and cluster variables.
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s introduced by . A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose union generates the algebra A and which satisfy various conditions.
Definitions
Suppose that F is an integral domain, such as the field Q(x1,...,xn) of rational functions in n variables over the rational numbers Q.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. The matrix is sometimes assumed to be skew symmetric, so that bx,y = –by,x. 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. It is common to picture a seed as a quiver
Quiver (mathematics)
In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation, V, of a quiver assigns a vector space V to each vertex x of the quiver and a linear map V to each...
with vertices the generating set, by drawing bx,y arrows from x to y if this number is positive. When bx,y is skew symmetrizable the quiver has no loops or 2-cycles.
A mutation of a seed, depending on a choice of vertex y of the cluster, is a new seed given by a generalization of tilting
Tilting theory
In algebra, tilting theory uses a tilting module T over an algebra A to construct tilting functors relating modules over A to modules over the tilted algebra EndA of endomorphisms of T....
as follows. Exchange the values of bx,y and by,x for all x in the cluster. If bx,y > 0 arrows and by,z > 0 then replace bx,z by bx,yby,z + bx,z. 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 inverse of a mutation is also a mutation: in other words, if A is a mutation of B, then B is a mutation of A.
A cluster algebra is constructed from a seed as follows.
If we repeatedly mutate the seed in all possible ways, we get a finite or infinite graph of seeds, where two seeds are joined if one can be obtained by mutating the other. The underlying algebra of the cluster algebra is the algebra generated by all the clusters of all the seeds in this graph. The cluster algebra also comes with the extra structure of the seeds of this graph.
A cluster algebra is said to be of finite type if it has only a finite number of seeds. showed that the cluster algebras of finite type can be classified in terms of the Dynkin diagrams of finite dimensional simple Lie algebras.
Cluster algebras of rank 1
If {x} is the cluster of a seed of rank 1, then the only mutation takes this to {2x–1}. So a cluster algebra of rank 1 is just a ring k[x,x–1] of Laurent polynomials, and it has just two clusters, {x} and {2x–1}. In particular it is of finite type and is associated with the Dynkin diagram A1.Cluster algebras of rank 2
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}. The variables are related byso are given by the sequence
which repeats with period 5. So this cluster algebra has exactly 5 clusters, and in particular is of finite type. It is associated with the Dynkin diagram A2.
There are similar examples with b12 = 1, –b21 = 2 or 3, where the analogous sequence of cluster variables repeats with period 6 or 8. These are also of finite type, and are associated with the Dynkin diagrams B2 and G2. However if |b12b21| ≥ 4 then the sequence of cluster variables is not periodic and the cluster algebra is of infinite type.
Cluster algebras of rank 3
Suppose we start with the quiver x1→x2→x3. Then the 14 clusters are:There are 6 cluster variables other than the 3 initial ones x1, x2, x3 given by
.They correspond to the 6 positive roots of the Dynkin diagram A3: more precisely the denominators are monomials in x1, x2, x3, corresponding to the expression of positive roots as the sum of simple roots.
The 3+6 cluster variables generate a cluster algebra of finite type, associated with the Dynkin diagram A3.
The 14 clusters are the vertices of the cluster graph, which is an associahedron
Associahedron
In mathematics, an associahedron or Stasheff polytope Kn is a convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a word of n letters and the edges correspond to single application of the associativity rule.Initially Jim Stasheff...
.
Grassmannians
Simple examples are given by the algebras of homogeneous functions on the GrassmannianGrassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
s. The Plücker coordinates provide some of the distinguished elements.
For the Grassmannian of planes in ℂn, the situation is even more simple. In that case, the Plücker coordinates provide all the distinguished elements and the clusters can be completely described using triangulation
Polygon triangulation
In computational geometry, polygon triangulation is the decomposition of a polygonal area P into a set of triangles, i.e., finding the set of triangles with pairwise non-intersecting interiors whose union is P....
s of a regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
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). One can distinguish between diagonals in the boundary, which belong to every cluster, and diagonals in the interior. This corresponds to a general distinction between coefficient variables and cluster variables.