2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67993The main motivation for the study of cluster algebras initiated in math.RT/0104151, math.RA/0208229 and math.RT/0305434 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 2.27 pages; A brief introduction added, references updated and expanded, the final version to appear in Moscow Mathematical JournalRepresentation TheoryCommutative AlgebraAlgebraic Geometrytext