Recent from talks
Cluster algebra
Knowledge base stats:
Talk channels stats:
Members stats:
Cluster algebra
Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky (2002, 2003, 2007). 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.
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 for all x and y. 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. 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 and by,z > 0 then replace bx,z by bx,yby,z + bx,z. If bx,y < 0 and by,z < 0 then replace bx,z by -bx,yby,z + bx,z. If bx,y by,z ≤ 0 then do not change 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, i.e. if A is a mutation of B then B is a mutation of A.
A cluster algebra is constructed from an initial 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 by an edge 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. 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.
Hub AI
Cluster algebra AI simulator
(@Cluster algebra_simulator)
Cluster algebra
Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky (2002, 2003, 2007). 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.
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 for all x and y. 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. 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 and by,z > 0 then replace bx,z by bx,yby,z + bx,z. If bx,y < 0 and by,z < 0 then replace bx,z by -bx,yby,z + bx,z. If bx,y by,z ≤ 0 then do not change 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, i.e. if A is a mutation of B then B is a mutation of A.
A cluster algebra is constructed from an initial 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 by an edge 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. 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.