Compare triangular bases of acyclic quantum cluster algebras

Qin, Fan

doi:10.1090/tran/7610

Compare triangular bases of acyclic quantum cluster algebras
HTML articles powered by AMS MathViewer

by Fan Qin PDF

Trans. Amer. Math. Soc. 372 (2019), 485-501 Request permission

Abstract:

Given a quantum cluster algebra, we show that its triangular bases defined by Berenstein and Zelevinsky and those defined by the author are the same for the seeds associated with acyclic quivers. This result implies that the Berenstein–Zelevinsky basis contains all of the quantum cluster monomials.

We also give an easy proof that the two bases are the same for the seeds associated with bipartite skew-symmetrizable matrices.

References

Arkady Berenstein and Andrei Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2005), no. 2, 405–455. MR 2146350, DOI 10.1016/j.aim.2004.08.003
Arkady Berenstein and Andrei Zelevinsky, Triangular bases in quantum cluster algebras, Int. Math. Res. Not. IMRN 6 (2014), 1651–1688. MR 3180605, DOI 10.1093/imrn/rns268
Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky, Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), no. 1, 59–119. MR 2480710, DOI 10.1007/s00029-008-0057-9
Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky, Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), no. 3, 749–790. MR 2629987, DOI 10.1090/S0894-0347-10-00662-4
Ming Ding and Fan Xu, Bases of the quantum cluster algebra of the Kronecker quiver, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 6, 1169–1178. MR 2916330, DOI 10.1007/s10114-011-0344-9
Vladimir V. Fock and Alexander B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 6, 865–930 (English, with English and French summaries). MR 2567745, DOI 10.24033/asens.2112
Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. MR 1887642, DOI 10.1090/S0894-0347-01-00385-X
Sergey Fomin and Andrei Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112–164. MR 2295199, DOI 10.1112/S0010437X06002521
Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 497–608. MR 3758151, DOI 10.1090/jams/890
Christof Geiß, Bernard Leclerc, and Jan Schröer, Kac-Moody groups and cluster algebras, Adv. Math. 228 (2011), no. 1, 329–433. MR 2822235, DOI 10.1016/j.aim.2011.05.011
Christof Geiss, Bernard Leclerc, and Jan Schröer, Generic bases for cluster algebras and the Chamber ansatz, J. Amer. Math. Soc. 25 (2012), no. 1, 21–76. MR 2833478, DOI 10.1090/S0894-0347-2011-00715-7
C. Geiß, B. Leclerc, and J. Schröer, Cluster structures on quantum coordinate rings, Selecta Math. (N.S.) 19 (2013), no. 2, 337–397. MR 3090232, DOI 10.1007/s00029-012-0099-x
David Hernandez and Bernard Leclerc, Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), no. 2, 265–341. MR 2682185, DOI 10.1215/00127094-2010-040
Masaki Kashiwara, Bases cristallines, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 6, 277–280 (French, with English summary). MR 1071626
Bernhard Keller, Cluster algebras, quiver representations and triangulated categories, Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 76–160. MR 2681708
Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-jin Oh, Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 349–426. MR 3758148, DOI 10.1090/jams/895
Yoshiyuki Kimura and Fan Qin, Graded quiver varieties, quantum cluster algebras and dual canonical basis, Adv. Math. 262 (2014), 261–312. MR 3228430, DOI 10.1016/j.aim.2014.05.014
Kyungyong Lee, Li Li, Dylan Rupel, and Andrei Zelevinsky, Greedy bases in rank 2 quantum cluster algebras, Proc. Natl. Acad. Sci. USA 111 (2014), no. 27, 9712–9716. MR 3263303, DOI 10.1073/pnas.1313078111
Kyungyong Lee, Li Li, and Andrei Zelevinsky, Greedy elements in rank 2 cluster algebras, Selecta Math. (N.S.) 20 (2014), no. 1, 57–82. MR 3147413, DOI 10.1007/s00029-012-0115-1
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
Gregg Musiker, Ralf Schiffler, and Lauren Williams, Bases for cluster algebras from surfaces, Compos. Math. 149 (2013), no. 2, 217–263. MR 3020308, DOI 10.1112/S0010437X12000450
Greg Muller, The existence of a maximal green sequence is not invariant under quiver mutation, Electron. J. Combin. 23 (2016), no. 2, Paper 2.47, 23. MR 3512669
Hiraku Nakajima, Quiver varieties and cluster algebras, Kyoto J. Math. 51 (2011), no. 1, 71–126. MR 2784748, DOI 10.1215/0023608X-2010-021
Pierre-Guy Plamondon, Cluster characters for cluster categories with infinite-dimensional morphism spaces, Adv. Math. 227 (2011), no. 1, 1–39. MR 2782186, DOI 10.1016/j.aim.2010.12.010
Fan Qin, $t$-analog of $q$-characters, bases of quantum cluster algebras, and a correction technique, Int. Math. Res. Not. IMRN 22 (2014), 6175–6232. MR 3283002, DOI 10.1093/imrn/rnt115
Fan Qin, Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. J. 166 (2017), no. 12, 2337–2442. MR 3694569, DOI 10.1215/00127094-2017-0006
Dylan Paul Thurston, Positive basis for surface skein algebras, Proc. Natl. Acad. Sci. USA 111 (2014), no. 27, 9725–9732. MR 3263305, DOI 10.1073/pnas.1313070111
Thao Tran, $F$-polynomials in quantum cluster algebras, Algebr. Represent. Theory 14 (2011), no. 6, 1025–1061. MR 2844755, DOI 10.1007/s10468-010-9226-6