ar X iv : m at h / 03 05 43 4 v 2 [ m at h . R T ] 5 J ul 2 00 3 CLUSTER ALGEBRAS III : UPPER BOUNDS AND DOUBLE BRUHAT CELLS
نویسنده
چکیده
We develop a new approach to cluster algebras based on the notion of an upper cluster algebra, defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [6], we show that, under an assumption of “acyclicity”, a cluster algebra coincides with its “upper” counterpart, and is finitely generated; in this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to an upper cluster algebra explicitly defined in terms of relevant combinatorial data.
منابع مشابه
ar X iv : m at h / 03 05 43 4 v 1 [ m at h . R T ] 3 0 M ay 2 00 3 CLUSTER ALGEBRAS III : UPPER BOUNDS AND DOUBLE BRUHAT CELLS
We develop a new approach to cluster algebras based on the notion of an upper cluster algebra, defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [6], we show that, under an assumption of “acyclicity”, a cluster algebra coincides with its “upper” counterpart, and is finitely generated; in this case, we also describe its defining ideal, an...
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