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Positivity and canonical bases in rank 2 cluster algebras of finite and affine types. (English) Zbl 1103.16018
Summary: The main motivation for the study of cluster algebras initiated by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497-529 (2002; Zbl 1021.16017)], B. L. Feń≠gin and A. Zelevinsky [Representations of Lie groups and Lie algebras, Proc. Summer Sch., Budapest 1971, Pt. 2, 25-77 (1985; Zbl 0599.17010)], and A. Berenstein, S. Fomin and A. Zelevinsky [Duke Math. J. 126, No. 1, 1-52 (2005; Zbl 1135.16013)] was to design an algebraic framework for understanding total positivity and canonical bases in semisimple algebraic groups. In this paper, we introduce and explicitly construct the canonical basis for a special family of cluster algebras of rank 2.

16S99 Associative rings and algebras arising under various constructions
20G05 Representation theory for linear algebraic groups
17B20 Simple, semisimple, reductive (super)algebras
05E15 Combinatorial aspects of groups and algebras (MSC2010)
22E46 Semisimple Lie groups and their representations
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