QUANTUM FRIEZE PATTERNS IN QUANTUM CLUSTER ALGEBRAS OF TYPE A

Year 2012, Volume: 12 Issue: 12, 103 - 115, 01.12.2012

Jean-philippe Burelle Grégoire Dupont

Abstract

We introduce a quantisation of the Coxeter-Conway frieze
patterns and prove that they realise quantum cluster variables in quantum
cluster algebras associated with linearly oriented Dynkin quivers of type A.
As an application, we obtain the explicit polynomials arising from the lower
bound phenomenon in these quantum cluster algebras.

Keywords

quantum cluster algebras, frieze patterns, continuant polynomials

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• Universit´e de Sherbrooke Boul. de l’universit´e, J1K 2R1 Sherbrooke QC, Canada. e-mail: [e-mail] Gr´egoire Dupont Institut de Math´ematiques de Jussieu – Paris Rive Gauche Universit´e Denis Diderot – Paris 7 rue du chevaleret Paris, France. e-mail: [e-mail]