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Cluster algebras are a class of commutative rings 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(''x''1,...,''x''''n'') of rational functions in ''n'' variables over the rational numbers Q. A cluster of rank ''n'' consists of a set of ''n'' elements of ''F'', usually assumed to be an algebraically independent set of generators of a field extension ''F''. A seed consists of a cluster of ''F'', together with an exchange matrix ''B'' with integer entries ''b''''x'',''y'' indexed by pairs of elements ''x'', ''y'' of the cluster. The matrix is sometimes assumed to be skew symmetric, so that ''b''''x'',''y'' = –''b''''y'',''x''. More generally the matrix might be skew symmetrizable, meaning there are positive integers ''d''''x'' associated with the elements of the cluster such that ''d''''x''''b''''x'',''y'' = –''d''''y''''b''''y'',''x''. It is common to picture a seed as a quiver with vertices the generating set, by drawing ''b''''x'',''y'' arrows from ''x'' to ''y'' if this number is positive. When ''b''''x'',''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 ''b''''x'',''y'' and ''b''''y'',''x'' for all ''x'' in the cluster. If ''b''''x'',''y'' > 0 and ''b''''y'',''z'' > 0 then replace ''b''''x'',''z'' by ''b''''x'',''y''''b''''y'',''z'' + ''b''''x'',''z''. If ''b''''x'',''y'' < 0 and ''b''''y'',''z'' < 0 then replace ''b''''x'',''z'' by -''b''''x'',''y''''b''''y'',''z'' + ''b''''x'',''z''. If ''b''''x'',''y'' ''b''''y'',''z'' <= 0 do not change ''b''''x'',''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 ''b''''t'',''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cluster algebra」の詳細全文を読む スポンサード リンク
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