A Correspondence between Rigid Modules Over Path Algebras and Simple Curves on Riemann Surfaces

References

• [Aigner 13] M. Aigner, Markov’s Theorem and 100 Years of the Uniqueness Conjecture, A Mathematical Journey from Irrational Numbers to Perfect Matchings. Cham: Springer, 2013.
   Google Scholar
• [Apruzzese and Igusa 18] P. J. Apruzzese and K. Igusa, Stability conditions for affine type A, arXiv:1804.09100.
   Google Scholar
• [Assem et al. 06] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory (London Mathematical Society Student Texts 65), Cambridge: Cambridge University Press, 2006.
   Google Scholar
• [Avramov and Buchweitz 00] L. L. Avramov and R.-O. Buchweitz, “Support Varieties and Cohomology Over Complete Intersections,” Invent. Math. 142 (2000), 285–318.
   Web of Science ®Google Scholar
• [Baumeister et al. 14] B. Baumeister, M. Dyer, C. Stump, and P. Wegener, “A Note on the Transitive Hurwitz Action on Decompositions of Parabolic Coxeter Elements,” Proc. Amer. Math. Soc. Ser. B. 1 (2014), 149–154.
   Google Scholar
• [Bessis 06] D. Bessis, “A Dual Braid Monoid for the Free Group,” J. Algebra. 302 (2006) 55–69.
   Web of Science ®Google Scholar
• [Brüstle and Zhang 11] T. Brüstle and J. Zhang. “ On the Cluster Category of a Marked Surface Without Punctures,” Algebra and Number Theory 5 (2011), 529–566.
   Web of Science ®Google Scholar
• [Caldero and Keller 06] P. Caldero and B. Keller, “From Triangulated Categories to Cluster Algebras II,” Ann. Sci. École Norm. Sup. 39:6 (2006), 983–1009.
   Web of Science ®Google Scholar
• [Canakci and Schroll 17] I. Canakci and S. Schroll, “Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces,” Adv. Math. 313 (2017), 1–49.
   Web of Science ®Google Scholar
• [Chavez 15] A. N. Chavez, “On the C-Vectors of an Acyclic Cluster Algebra,” Int. Math. Res. Not. 6 (2015), 1590–1600.
   Web of Science ®Google Scholar
• [Crawley-Boevey 92] W. Crawley-Boevey, “Exceptional sequences of representations of quivers,” Proceedings of the Sixth International Conference on Representations of Algebras (Ottawa, ON, 1992), 7 pp., Carleton-Ottawa Math. Lecture Note Ser., 14, Carleton University, Ottawa, ON, 1992.
   Google Scholar
• [Davison 18] B. Davison, “Positivity for Quantum Cluster Algebras,” Ann. of Math. 187:1 (2018), 157–219.
   Web of Science ®Google Scholar
• [Davison et al. 15] B. Davison, D. Maulik, J. Schürmann, and B. Szendröi, “Purity for Graded Potentials and Quantum Cluster Positivity,” Compos. Math. 151:10 (2015), 1913–1944.
   Web of Science ®Google Scholar
• [Felikson and Tumarkin 17] A. Felikson and P. Tumarkin, “Acyclic Cluster Algebras, Reflections Groups, and Curves on a Punctured Disc,” 2017. arXiv:1709.10360.
   Google Scholar
• [Fomin and Zelevinsky 02] S. Fomin and A. Zelevinsky, “Cluster Algebras I: Foundations,” J. Amer. Math. Soc. 15:2 (2002), 497–529.
   Web of Science ®Google Scholar
• [Fomin and Zelevinsky 07] S. Fomin and A. Zelevinsky, “Cluster Algebras IV: Coefficients,” Compos. Math. 143 (2007), 112–164.
   Web of Science ®Google Scholar
• [Gross et al. 18] M. Gross, P. Hacking, S. Keel, and M. Kontsevich, “Canonical Bases for Cluster Algebras,” J. Amer. Math. Soc. 31:2 (2018), 497–608.
   Web of Science ®Google Scholar
• [Hubery and Krause 16] A. Hubery and H. Krause, “A Categorification of Non-Crossing Partitions,” J. Eur. Math. Soc. 18:10 (2016), 2273–2313.
   Web of Science ®Google Scholar
• [Igusa and Schiffler 10] K. Igusa and R. Schiffler, “Exceptional Sequences and Clusters,” J. Algebra 323:8 (2010), 2183–2202.
   Web of Science ®Google Scholar
• [Kontsevich 95] M. Kontsevich, “Homological Algebra of Mirror Symmetry,” Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, pp. 120–139.
   Google Scholar
• [Lee and Schiffler 13] K. Lee and R. Schiffler, “Positivity for Cluster Algebras of Rank 3,” Publ. Res. Inst. Math. Sci. 49:3 (2013), 601–649.
   Web of Science ®Google Scholar
• [Lee and Schiffler 15] K. Lee and R. Schiffler, “Positivity for Cluster Algebras,” Ann. of Math. 182:1 (2015), 73–125.
   Web of Science ®Google Scholar
• [Musiker et al. 11] G. Musiker, R. Schiffler, and L. Williams, “Positivity for Cluster Algebras from Surfaces,” Adv. Math. 227 (2011), 2241–2308.
   Web of Science ®Google Scholar
• [Schofield 92] A. Schofield, “General Representations of Quivers,” Proc. London Math. Soc. 65:1 (1992), 46–64.
   Google Scholar
• [Seven 15] A. Seven, “Cluster Algebras and Symmetric Matrices,” Proc. Amer. Math. Soc. 143 (2015), 469–478.
   Web of Science ®Google Scholar
• [Shende 15] V. Shende, D. Treumann, H. Williams, and E. Zaslow, “Cluster Varieties from Legendrian Knots,” 2015, arXiv:1512.08942.
   Google Scholar
• [Shende et al. 16] V. Shende, D. Treumann and H. Williams, On the Combinatorics of Exact Lagrangian Surfaces,” 2016, arXiv:1603.07449.
   Google Scholar
• [Speyer and Thomas 13] D. Speyer and H. Thomas, “Acyclic Cluster Algebras Revisited,” In Algebras, Quivers and Representations, vol. 8, pp. 275–298, Abel Symp., Springer, Heidelberg, 2013.
   Google Scholar
• [Treumann 18] D. Treumann, H. Williams and E. Zaslow, “Kasteleyn operators from mirror symmetry,” arXiv:1810.05985.
   Google Scholar
• [Zhang et al. 13] J. Zhang, Y. Zhou and B. Zhu, “Cotorsion Pairs in the Cluster Category of a Marked Surface,” J. Algebra 391 (2013), 209–226.
   Web of Science ®Google Scholar