Cluster algebra

In mathematics , cluster algebras are used in representation theory , low-dimensional topology and higher Teichmüller theory . Cluster algebras are subalgebras of given by producers who membered n-in "clusters" are combined with by skew symmetric - exchange matrices given transition rules (so-called. Mutations ) between clusters. ${\displaystyle \mathbb {Q} (x_{1},\ldots ,x_{n})}$ ${\displaystyle n\times n}$

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

definition

A cluster is made up of a pair ${\displaystyle (x,B)}$

• an n-tuple of algebraically independent variables, ${\displaystyle x=(x_{1},\ldots ,x_{n})}$
• a skew-symmetric, integer matrix , the exchange matrix . ${\displaystyle n\times n}$ ${\displaystyle B=(b_{ij})}$

For the mutation is defined by with ${\displaystyle k=1,\ldots ,n}$ ${\displaystyle \mu _{k}}$ ${\displaystyle \mu _{k}(x,B)=({\tilde {x}},{\tilde {B}})}$

${\displaystyle {\tilde {x}}_{i}=x_{i}}$ For ${\displaystyle i\neq k}$

${\displaystyle {\tilde {x}}_{k}={\frac {1}{x_{k}}}\left(\prod _{\{j\colon b_{jk}>0\}}x_{j}^{b_{jk}}+\prod _{\{j\colon b_{jk}<0\}}x_{j}^{-b_{jk}}\right)}$

${\displaystyle {\tilde {b}}_{ij}=b_{ij}+{\frac {|b_{ik}|b_{kj}+b_{ik}|b_{kj}|}{2}}}$if ${\displaystyle i\neq k,j\neq k}$

${\displaystyle {\tilde {b}}_{kj}=-b_{kj},{\tilde {b}}_{ik}=-b_{ik}}$.

${\displaystyle \mu _{k}(x,B)}$is also a cluster, are involutions . ${\displaystyle \mu _{1},\ldots ,\mu _{n}}$

A cluster algebra is created from a cluster by iterated application of all possible mutations . Cluster algebra is called of finite type if there are only finitely many clusters. ${\displaystyle \mu _{1},\ldots ,\mu _{n}}$

Examples

A1

For must be the asymmetric matrix , one calculates ${\displaystyle n=1}$ ${\displaystyle B=0}$

${\displaystyle \mu _{1}((x_{1}),0)=(({\frac {2}{x_{1}}}),0)}$.

Because of this, this is a finite-type cluster algebra, it corresponds to the Cartan matrix . ${\displaystyle \mu _{1}^{2}=id}$ ${\displaystyle A_{1}}$

A2

Be and . One calculates ${\displaystyle n=2}$ ${\displaystyle b_{12}=-b_{21}=1}$

${\displaystyle \mu _{1}((x_{1},x_{2}),B)=(({\frac {1+x_{2}}{x_{1}}},x_{2}),-B),\mu _{2}(x_{1},x_{2})=((x_{1},{\frac {1+x_{2}}{x_{1}}}),-B),}$

${\displaystyle \mu _{2}\mu _{1}((x_{1},x_{2}),B)=(({\frac {1+x_{2}}{x_{1}}}),{\frac {1+x_{1}+x_{2}}{x_{1}x_{2}}}),B),\mu _{1}\mu _{2}((x_{1},x_{2}),B)=(({\frac {1+x_{1}+x_{2}}{x_{1}x_{2}}},{\frac {1+x_{1}}{x_{2}}}),B),}$

${\displaystyle \mu _{1}\mu _{2}\mu _{1}((x_{1},x_{2}),B)=(({\frac {1+x_{1}}{x_{2}}},{\frac {1+x_{1}+x_{2}}{x_{1}x_{2}}}),-B),\mu _{2}\mu _{1}\mu _{2}((x_{1},x_{2}),B)=(({\frac {1+x_{1}+x_{2}}{x_{1}x_{2}}},{\frac {1+x_{2}}{x_{1}}}),-B),}$

${\displaystyle \mu _{2}\mu _{1}\mu _{2}\mu _{1}((x_{1},x_{2}),B)=(({\frac {1+x_{1}}{x_{2}}},x_{1}),B),\mu _{1}\mu _{2},\mu _{1}\mu _{2}((x_{1},x_{2}),B)=((x_{2},{\frac {1+x_{2}}{x_{1}}}),B),}$

${\displaystyle \mu _{1}\mu _{2}\mu _{1}\mu _{2}\mu _{1}((x_{1},x_{2}),B)=((x_{2},x_{1}),-B),\mu _{2}\mu _{1}\mu _{2},\mu _{1}\mu _{2}((x_{1},x_{2}),B)=((x_{2},x_{1}),-B),}$

${\displaystyle \ldots }$

${\displaystyle \mu _{2}\mu _{1}\mu _{2},\mu _{1}\mu _{2}\mu _{1}\mu _{2}\mu _{1}\mu _{2}\mu _{1}((x_{1},x_{2}),B)=((x_{1},x_{2}),B),\mu _{1}\mu _{2}\mu _{1}\mu _{2}\mu _{1}\mu _{2}\mu _{1}\mu _{2},\mu _{1}\mu _{2}((x_{1},x_{2}),B)=((x_{1},x_{2}),B).}$

This cluster algebra is therefore of finite type, it corresponds to the Cartan matrix . ${\displaystyle A_{2}}$

For and one obtains cluster algebras of infinite type. ${\displaystyle n=2}$ ${\displaystyle \mid b_{12}\mid \geq 2}$

Cluster algebras of topological origin

Triangulations and their associated exchange matrices.

A cluster algebra is assigned to a triangulated oriented surface as follows: ${\displaystyle (x,B)}$

• the variables are the edges of the triangulation,
• ${\displaystyle b_{ij}=1}$if the i-th and j-th edges follow one another clockwise within a triangle,
• ${\displaystyle b_{ij}=-1}$if the j-th and i-th edges follow one another within a clockwise triangle,
• ${\displaystyle b_{ij}=0}$ otherwise.

More generally, cluster algebras can also be associated with surfaces broken down into (possibly degenerate) triangles (see the work of Fomin-Shapiro-Thurston); the cluster algebras obtained in this way are called cluster algebras of topological origin.

Mutations correspond to flips of the triangulation: one edge is replaced by the complementary diagonal.

The mutations in this case are given by flips of the edges of the triangulation, i. H. to one edge one looks at the square spanned by the two adjacent triangles and then replaces the edge with the other diagonal of this square.

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 mutation-finite type are either cluster algebras of topological origin or are equivalent to one of 11 exception algebras. Mutational finiteness is more general than finite type.

literature

• Fomin, Sergey; Zelevinsky, Andrei: Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2002), no. 2, 497-529 pdf
• Fomin, Sergey; Zelevinsky, Andrei: Cluster algebras. II. Finite type classification. Invent. Math. 154 (2003), no. 1, 63-121. pdf
• Fomin, Sergey; Shapiro, Michael; Thurston, Dylan: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201 (2008), no. 1, 83-146. pdf
• Felikson, Anna; Shapiro, Michael; Tumarkin, Pavel: Skew-symmetric cluster algebras of finite mutation type. J. Eur. Math. Soc. (JEMS) 14 (2012), no. 4, 1135-1180. pdf

Web links

• What is a cluster algebra? (2007; PDF; 103 kB)
• Fomin, Sergey: Total positivity and cluster algebras. (PDF; 332 kB) Proceedings of the International Congress of Mathematicians. Volume II, 125-145, Hindustan Book Agency, New Delhi, 2010.
• Leclerc, Bernard Cluster algebras and representation theory. (PDF; 257 kB) Proceedings of the International Congress of Mathematicians. Volume IV, 2471-2488, Hindustan Book Agency, New Delhi, 2010
• Williams, Lauren: Cluster algebras: an introduction. (PDF) Bulletin of the American Mathematical Society 51, no. 1, 1–26 (2014).
• Cluster algebras portal