Editing Cluster algebra

'''Cluster algebras''' are a class of [[commutative ring]]s introduced by {{harvs|txt|last=Fomin|author1-link=Sergey Fomin|last2=Zelevinsky|author2-link=Andrei Zelevinsky|year=2002|year2=2003|year3=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 over a field|algebra]] ''A'' and which satisfy various conditions.

==Definitions==

Suppose that ''F'' is an [[integral domain]], such as the [[Field (mathematics)|field]] '''Q'''(''x''<sub>1</sub>,...,''x''<sub>''n''</sub>) of [[rational function]]s in ''n'' variables over the [[rational number]]s '''Q'''.

A '''cluster''' of '''rank''' ''n'' consists of a set of ''n'' elements {''x'', ''y'', ...} of ''F'', usually assumed to be an [[Algebraic independence|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 ''b''<sub>''x'',''y''</sub> indexed by pairs of elements ''x'', ''y'' of the cluster. The matrix is sometimes assumed to be [[Skew-symmetric matrix|skew-symmetric]], so that ''b''<sub>''x'',''y''</sub> = –''b''<sub>''y'',''x''</sub> for all ''x'' and ''y''. More generally the matrix might be skew-symmetrizable, meaning there are positive integers ''d''<sub>''x''</sub> associated with the elements of the cluster such that ''d''<sub>''x''</sub>''b''<sub>''x'',''y''</sub> = –''d''<sub>''y''</sub>''b''<sub>''y'',''x''</sub> for all ''x'' and ''y''. It is common to picture a seed as a [[quiver (mathematics)|quiver]] whose vertices are the generating set, by drawing ''b''<sub>''x'',''y''</sub> arrows from ''x'' to ''y'' if this number is positive. When ''b''<sub>''x'',''y''</sub> 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 theory|tilting]] as follows. Exchange the values of ''b''<sub>''x'',''y''</sub> and ''b''<sub>''y'',''x''</sub> for all ''x'' in the cluster. If  ''b''<sub>''x'',''y''</sub> > 0 and ''b''<sub>''y'',''z''</sub> > 0  then replace ''b''<sub>''x'',''z''</sub> by  ''b''<sub>''x'',''y''</sub>''b''<sub>''y'',''z''</sub> + ''b''<sub>''x'',''z''</sub>. If  ''b''<sub>''x'',''y''</sub> < 0 and ''b''<sub>''y'',''z''</sub> < 0  then replace ''b''<sub>''x'',''z''</sub> by  -''b''<sub>''x'',''y''</sub>''b''<sub>''y'',''z''</sub> + ''b''<sub>''x'',''z''</sub>. If ''b''<sub>''x'',''y''</sub> ''b''<sub>''y'',''z''</sub> ≤ 0 then do not change ''b''<sub>''x'',''z''</sub>. Finally replace ''y'' by a new generator ''w'', where 
:<math>wy=\prod_{t: \, b_{t,y}>0}t^{b_{t,y}} + \prod_{t: \, b_{t,y}<0}t^{-b_{t,y}}</math>
where the products run through the elements ''t'' in the cluster of the seed such that ''b''<sub>''t'',''y''</sub> 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 (discrete mathematics)|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. {{harvtxt|Fomin|Zelevinsky|2003}} showed that the cluster algebras of finite type can be classified in terms of the  [[Dynkin diagram]]s of finite-dimensional [[simple Lie algebra]]s.

==Examples==

===Cluster algebras of rank 1===

If {''x''} is the cluster of a seed of rank 1, then the only mutation takes this to {2''x''<sup>−1</sup>}. So a cluster algebra of rank 1 is just a ring ''k''[''x'',''x''<sup>−1</sup>] of [[Laurent polynomial]]s, and it has just two clusters, {''x''} and {2''x''<sup>−1</sup>}. In particular it is of finite type and is associated with the Dynkin diagram A<sub>1</sub>.

===Cluster algebras of rank 2===

Suppose that we start with the cluster {''x''<sub>1</sub>, ''x''<sub>2</sub>} and take the exchange matrix with ''b''<sub>12</sub> = –b<sub>21</sub> = 1. Then mutation gives a sequence of variables ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ''x''<sub>4</sub>,... such that the clusters are given by adjacent pairs {''x''<sub>''n''</sub>, ''x''<sub>''n''+1</sub>}. The variables are related by 
:<math>\displaystyle x_{n-1}x_{n+1}=1+x_n,</math>
so are given by the sequence

:<math>x_1, \ x_2, \ x_3 = \frac{1+x_2}{x_1}, \ x_4 = \frac{1+x_3}{x_2}=\frac{1+x_1+x_2}{x_1x_2},</math>
:<math>x_5 = \frac{1+x_4}{x_3} = \frac{1+x_1}{x_2}, \ x_6 = \frac{1+x_5}{x_4} = x_1, \ x_7 = \frac{1+x_6}{x_5} = x_2, \ \ldots</math>
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 A<sub>2</sub>.

There are similar examples with ''b''<sub>12</sub> = 1, –''b''<sub>21</sub> = 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 B<sub>2</sub> and G<sub>2</sub>. However if |''b''<sub>12</sub>''b''<sub>21</sub>| ≥ 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 ''x''<sub>1</sub> → ''x''<sub>2</sub> → ''x''<sub>3</sub>. Then the 14 clusters are:
:<math>\left\{ x_1,x_2,x_3 \right\},</math>
:<math>\left\{\frac{1+x_2}{x_1},x_2,x_3 \right\},</math>
:<math>\left\{x_1, \frac{x_1 + x_3}{x_2},x_3 \right\},</math> 
:<math>\left\{x_1,x_2,\frac{1+x_2}{x_3}\right\},</math>
:<math>\left\{\frac{1+x_2}{x_1}, \frac{x_1 +(1+x_2)x_3}{x_1 x_2},x_3 \right\},</math>
:<math>\left\{\frac{1+x_2}{x_1},x_2,\frac{1+x_2}{x_3} \right\},</math>
:<math>\left\{\frac{x_1+(1+x_2)x_3}{x_1x_2},\frac{x_1 + x_3}{x_2},x_3 \right\},</math>
:<math>\left\{x_1,\frac{x_1+x_3}{x_2},\frac{(1+x_2)x_1+x_3}{x_2x_3} \right\},</math> 
:<math>\left\{x_1,\frac{(1+x_2)x_1 + x_3}{x_2 x_3},\frac{1+x_2}{x_3} \right\},</math>
:<math>\left\{\frac{1+x_2}{x_1},\frac{x_1+(1+x_2)x_3}{x_1 x_2},\frac{(1+x_2)x_1 +(1+x_2)x_3}{x_1 x_2x_3}\right\},</math>
:<math>\left\{\frac{1+x_2}{x_1},\frac{(1+x_2)x_1 +(1+x_2)x_3}{x_1 x_2x_3},
\frac{1+x_2}{x_3} \right\},</math>
:<math>\left\{\frac{x_1+(1+x_2)x_3}{x_1x_2},\frac{x_1+x_3}{x_2},\frac{(1+x_2)x_1+(1+x_2)x_3}{x_1 x_2 x_3} \right\},</math>
:<math>\left\{\frac{(1+x_2)x_1 +(1+x_2)x_3}{x_1 x_2x_3},\frac{x_1+x_3}{x_2},\frac{(1+x_2)x_1+x_3}{x_2 x_3} \right\},</math> 
:<math>\left\{\frac{(1+x_2)x_1+(1+x_2)x_3}{x_1 x_2 x_3},\frac{(1+x_2)x_1+x_3}{x_2 x_3},\frac{1+x_2}{x_3} \right\}.</math>

There are 6 cluster variables other than the 3 initial ones ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub> given by
:<math>\frac{1+x_2}{x_1},\frac{x_1 + x_3}{x_2},\frac{1+x_2}{x_3}, \frac{x_1+(1+x_2)x_3}{x_1x_2}, \frac{(1+x_2)x_1+x_3}{x_2 x_3}, \frac{(1+x_2)x_1 +(1+x_2)x_3}{x_1 x_2x_3}</math>. 
They correspond to the 6 positive roots of the Dynkin diagram A<sub>3</sub>: more precisely the denominators are monomials in ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, 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 A<sub>3</sub>.
The 14 clusters are the vertices of the cluster graph, which is an [[associahedron]].

===Grassmannians===
Simple examples are given by the algebras of homogeneous functions on the [[Grassmannian]]s. The [[Plücker coordinates]] provide some of the distinguished elements.

[[File:Cluster algebra.svg|thumb|300px|Mutation between two triangulations of the heptagon]]
For the Grassmannian of planes in ℂ<sup>''n''</sup>, 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 [[Polygon triangulation|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.

===Cluster algebras arising from surfaces===
Suppose '''S''' is a [[Compact space|compact]] [[Connected space|connected]] [[Orientability|oriented]] [[Riemann surface]] and '''M''' is a [[Empty set|non-empty]] finite set of points in '''S''' that contains at least one point from each [[Boundary (topology)|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''. It has been shown by Fomin-Shapiro-Thurston that if '''S''' is not a closed surface, or if '''M''' has more than one point, then the (tagged) arcs on ('''S''', '''M''') parameterize the set of cluster variables of certain cluster algebra ''A''('''S''', '''M'''), which depends only on ('''S''', '''M''') and the choice of some coefficient system, in such a way that the set of (tagged) triangulations of ('''S''', '''M''') is in one-to-one correspondence with the set of clusters of ''A''('''S''', '''M'''), two (tagged) triangulations being related by a ''flip'' if and only if the clusters they correspond to are related by cluster mutation.

===Double Bruhat Cells===

For <math>G</math> a [[reductive group]] such as <math>GL_n</math> with [[Borel subgroup]]s <math>B_\pm</math> then on <math>G^{u,v} = B u B \cap B_- v B_-</math> (where <math>u</math> and <math>v</math> are in the [[Weyl group]]) there are cluster coordinate charts depending on reduced word decompositions of <math>u</math> and <math>v</math>. These are called factorization parameters and their structure is encoded in a wiring diagram. With only <math>B</math> or only <math>B_-</math>, this is [[Bruhat decomposition]].

==References==
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==External links==
*Fomin's [http://www.math.lsa.umich.edu/~fomin/cluster.html Cluster algebra portal]
*[http://www.math.lsa.umich.edu/~fomin/papers.html Fomin's papers on cluster algebras]

[[Category:Algebras]]
[[Category:Commutative algebra]]