QUANTUM FRIEZE PATTERNS IN QUANTUM CLUSTER ALGEBRAS OF TYPE A

Yıl 2012, Cilt: 12 Sayı: 12, 103 - 115, 01.12.2012

Jean-philippe Burelle Grégoire Dupont

Öz

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.

Anahtar Kelimeler

quantum cluster algebras, frieze patterns, continuant polynomials

Kaynakça

• I. Assem and G. Dupont, Friezes and a construction of the euclidean cluster variables, J. Pure Appl. Algebra, 215 (2011), 2322–2340.
• I. Assem, G. Dupont, R. Schiffler, and D. Smith, Friezes, strings and cluster variables, Glasg. Math. J., 54(1) (2012), 27–60.
• I. Assem, C. Reutenauer, and D. Smith, Friezes, Adv. Math., 225(6) (2010), –3165.
• K. Baur and R. Marsh, Frieze patterns for punctured discs, J. Algebraic Com- bin., 30(3) (2009), 349–379.
• A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J., 126(1) (2005), 1–52.
• A. Berenstein and A. Zelevinsky, Quantum cluster algebras, Adv. Math., 195(2) (2005), 405–455.
• F. Bergeron and C. Reutenauer, SLk-tilings of the plane, Illinois J. Math., (1) (2010), 263–300.
• A. Buan, R. Marsh, M. Reineke, I. Reiten, and G. Todorov, Tilting theory and cluster combinatorics, Adv. Math., 204(2) (2006), 572–618.
• P. Caldero and F. Chapoton, Cluster algebras as Hall algebras of quiver rep- resentations, Comment. Math. Helv., 81 (2006), 596–616.
• P. Caldero, F. Chapoton, and R. Schiffler, Quivers with relations arising from clusters (Ancase), Trans. Amer. Math. Soc., 358(3) (2006), 1347–1354.
• H. S. M. Coxeter, Frieze patterns, Acta Arith., 18 (1971), 297–310.
• G. Dupont, Quantized Chebyshev polynomials and cluster characters with co- efficients, J. Algebraic Combin., 31(4) (2010), 501–532. , Cluster multiplication in regular components via generalized Chebyshev polynomials, Algebr. Represent. Theory, 15(3) (2012), 527–549. , Positivity for regular cluster characters in a cyclic cluster algebras, J. Algebra Appl., to appear.
• C. Geiss, B. Leclerc, and J. Schr¨oer, Cluster structures on quantum coordinate rings, to appear in Selecta Mathematica. D. Happel, Triangulated categories in the representation theory of finite di- mensional algebras, London Mathematical Society Lecture Note Series, 119, Cambridge University Press, Cambridge, 1988.
• D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, arXiv:1109.0862v1 [math.QA], (2011).
• B. Keller, Cluster algebras, quiver representations and triangulated categories, Triangulated categories, London Math. Soc. Lecture Note Ser., 375, Cambridge Univ. Press, Cambridge, (2010), 76–160.
• Y. Kimura, Quantum unipotent subgroup and dual canonical basis, Kyoto J. Math., 52(2) (2012), 277–331.
• P. Lampe, A quantum cluster algebra of Kronecker type and the dual canonical basis, Int. Math. Res. Not. IMRN, (2011), no. 13, 2970–3005. , Quantum cluster algebras of type A and the dual canonical basis, arXiv:1101.0580v1 [math.RT] (2011).
• J. Propp, The combinatorics of frieze patterns and Markoff numbers, arXiv:math/0511633v4 [math.CO], (2008).
• R. Schiffler and H. Thomas, On cluster algebras arising from unpunctured surfaces, Int. Math. Res. Not. IMRN, (2009), no. 17, 3160–3189. Jean-Philippe Burelle
• 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]