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Cluster algebras. I: Foundations. (English) Zbl 1021.16017
The authors introduce a new class of algebras called “cluster algebras”, and exhibit the cluster algebra structure for some well-known algebras such as \(\mathbb C[\text{SL}_3/N]\) (\(N\) being the subgroup of \(\text{SL}_3\) consisting of all unipotent upper triangular matrices), \(\mathbb C[\text{Gr}_{2,n+3}]\) (\(\text{Gr}_{2,n+3}\) being the Grassmannian of 2-dimensional subspaces of \(\mathbb C^{n+3}\)). After introducing the cluster algebras, the authors first derive some structural properties of these algebras, then study the cluster algebras of rank 2. The authors also conjecture that for a complex semisimple, simply connected algebraic group \(G\), \(\mathbb C[G]\) and \(\mathbb C[G/N]\) are cluster algebras.

MSC:
16S34 Group rings
13F60 Cluster algebras
16S50 Endomorphism rings; matrix rings
17B20 Simple, semisimple, reductive (super)algebras
20G05 Representation theory for linear algebraic groups
14M15 Grassmannians, Schubert varieties, flag manifolds
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