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Geometric description of C-vectors and real Lösungen

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Abstract

We introduce real Lösungen as an analogue of real roots. For each mutation sequence of an arbitrary skew-symmetrizable matrix, we define a family of reflections along with associated vectors which are real Lösungen and a set of curves on a Riemann surface. The matrix consisting of these vectors is called L-matrix. We explain how the L-matrix naturally arises in connection with the C-matrix. Then we conjecture that the L-matrix depends (up to signs of row vectors) only on the seed, and that the curves can be drawn without self-intersections, providing a new combinatorial/geometric description of c-vectors.

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Notes

  1. Some authors call them quasi-Cartan matrices. For example, see [3].

  2. Historically, when Killing investigated the structure of a finite dimensional simple Lie algebra L with Cartan subalgebra \({\mathfrak {h}}\), the roots of the characteristic polynomial \(\det ({\text {ad}} _{L}x-t)\), \(x \in {\mathfrak {h}}\), were called the roots [8].

  3. An alternative geometric model can be found in [12].

References

  1. Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras, Vol. 1, London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  2. Barot, M., Marsh, R.J.: Reflection group presentations arising from cluster algebras. Trans. Am. Math. Soc. 367, 1945–1967 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barot, M., Rivera, D.: Generalized Serre relations for Lie algebras associated with positive unit forms. J. Pure Appl. Algebra 211(2), 360–373 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barot, M., Kussin, D., Lenzing, H.: The Lie algebra associated to a unit form. J. Algebra 296(1), 1–17 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baumeister, B., Dyer, M., Stump, C., Wegener, P.: A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements. Proc. Am. Math. Soc. Ser. B 1, 149–154 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benkart, G., Zelmanov, E.: Lie algebras graded by finite root systems and intersection matrix algebras. Invent. Math. 126(1), 1–45 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berman, S., Moody, R.V.: Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy. Invent. Math. 108(2), 323–347 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bourbaki, N.: Elements of the History of Mathematics. Springer, Berlin (1994)

    MATH  Google Scholar 

  9. Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations I: mutations. Selecta Math. 14(1), 59–119 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations II: applications to cluster algebras. J. Am. Math. Soc. 23(3), 749–790 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Felikson, A., Tumarkin, P.: Coxeter groups and their quotients arising from cluster algebras. Int. Math. Res. Not. IMRN 2016(17), 5135–5186 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Felikson, A., Tumarkin, P.: Acyclic cluster algebras, reflections groups, and curves on a punctured disc. Adv. Math. 340, 855–882 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fomin, S., Zelevinsky, A.: Cluster algebras IV: coefficients. Compos. Math. 143, 112–164 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gross, M., Hacking, P., Keel, S., Kontsevich, M.: Canonical bases for cluster algebras. J. Am. Math. Soc. 31(2), 497–608 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gupta, M.: A formula for \(F\)-polynomials in terms of \(C\)-cectors and stabilization of \(F\)-polynomials, preprint. arXiv:1812.01910

  16. Hubery, A., Krause, H.: A categorification of non-crossing partitions. J. Eur. Math. Soc. 18(10), 2273–2313 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Igusa, K., Schiffler, R.: Exceptional sequences and clusters. J. Algebra 323(8), 2183–2202 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kac, V.G.: Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56, 57–92 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kac, V.G.: Infinite Dimensional Lie Algebras. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  20. Lee, K.-H., Lee, K.: A correspondence between rigid modules over path algebras and simple curves on Riemann surfaces, to appear in Exp. Math. arXiv:1703.09113

  21. Nájera Chávez, N.: On the c-vectors of an acyclic cluster algebra. Int. Math. Res. Not. IMRN 6, 1590–1600 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nakanishi, T., Zelevinsky, A.: On tropical dualities in cluster algebras. Contemp. Math. 565, 217–226 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nicolai, H., Fischbacher, T.: Low level representations for \(E_{10}\) and \(E_{11}\). Contemp. Math. 343, 191–227 (2004)

    Article  MATH  Google Scholar 

  24. Plamondon, P.-G.: Cluster algebras via cluster categories with infinite-dimensional morphism spaces. Compos. Math. 147, 1921–1954 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ringel, C.M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics, vol. 1099. Springer, Berlin (1984)

    MATH  Google Scholar 

  26. Saito, K., Yoshii, D.: Extended affine root system. IV. Simply-laced elliptic Lie algebras. Publ. Res. Inst. Math. Sci. 36(3), 385–421 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schofield, A.: General representations of quivers. Proc. Lond. Math. Soc. (3) 65(1), 46–64 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Seven, A.: Mutation classes of skew-symmetrizable \(3 \times 3\) matrices. Proc. Am. Math. Soc. 141, 1493–1504 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Seven, A.: Cluster algebras and symmetric matrices. Proc. Am. Math. Soc. 143, 469–478 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Seven, A.: Reflection group relations arising from cluster algebras. Proc. Am. Math. Soc. 144, 4641–4650 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Seven, A.: Cluster algebras and symmetrizable matrices. Proc. Am. Math. Soc. 147, 2809–2814 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Slodowy, P.: Singularitäten, Kac-Moody Lie-Algebren, assoziierte Gruppen und Verallgemeinerungen. Universität Bonn, Habilitationsschrift (1984)

    Google Scholar 

  33. Slodowy, P.: Beyond Kac–Moody algebras and inside. Can. Math. Soc. Conf. Proc. 5, 361–371 (1986)

    MathSciNet  MATH  Google Scholar 

  34. Speyer, D., Thomas, H.: Acyclic cluster algebras revisited, Algebras, quivers and representations. In: Abel Symp. vol. 8. Springer, Heidelberg, pp. 275–298 (2013)

  35. Xia, L.-M., Hu, N.: A class of Lie algebras arising from intersection matrices. Front. Math. China 10(1), 185–198 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are very grateful to Pavel Tumarkin, Ahmet Seven and anonymous referees for correspondences and comments, which substantially improved the exposition of this paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Kyu-Hwan Lee

  2. Department of Mathematics, University of Alabama, Tuscaloosa, AL, 35487, USA

    Kyungyong Lee

  3. Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea

    Kyungyong Lee

  4. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Matthew R. Mills

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  1. Kyu-Hwan Lee
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Correspondence to Kyu-Hwan Lee.

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K.-H. Lee: This work was partially supported by a grant from the Simons Foundation (#712100). K. Lee: This work was partially supported by NSF Grant DMS 2042786, the University of Alabama, and Korea Institute for Advanced Study. M. R. Mills: This material is based upon work supported by the National Science Foundation under Award no. 1803521 and Michigan State University.

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Lee, KH., Lee, K. & Mills, M.R. Geometric description of C-vectors and real Lösungen. Math. Z. 303, 44 (2023). https://doi.org/10.1007/s00209-022-03180-8

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