Advertisement

Journal of Algebraic Combinatorics

, Volume 44, Issue 1, pp 223–247 | Cite as

Cylindrical Dyck paths and the Mazorchuk–Turowska equation

  • Jonas T. Hartwig
  • Daniele Rosso
Article

Abstract

We classify all solutions (pq) to the equation \(p(u)q(u)=p(u+\beta )q(u+\alpha )\) where p and q are complex polynomials in one indeterminate u, and \(\alpha \) and \(\beta \) are fixed but arbitrary complex numbers. This equation is a special case of a system of equations which ensures that certain algebras defined by generators and relations are non-trivial. We first give a necessary condition for the existence of non-trivial solutions to the equation. Then, under this condition, we use combinatorics of generalized Dyck paths to describe all solutions and a canonical way to factor each solution into a product of irreducible solutions.

Keywords

Dyck path Generalized Weyl algebra 

Mathematics Subject Classification

05A10 16S35 

References

  1. 1.
    Bavula, V.V.: Generalized Weyl algebras and their representations. Algebra i Analiz 4(1), 75–97 (1992); English translation in St. Petersburg. Math. J. 4, 71–92 (1993)Google Scholar
  2. 2.
    Bavula, V., Bekkert, V.: Indecomposable representations of generalized Weyl algebras. Commun. Algebra 28, 5067–5100 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Farinati, M.A., Solotar, A., Suárez-Álvarez, M.: Hochschild homology and cohomology of generalized Weyl algebras. Ann. Inst. Fourier (Grenoble) 53(2), 465–488 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fukukawa, Y.: Counting generalized Dyck paths, arXiv:1304.5595 [math.CO]
  5. 5.
    Futorny, V., Hartwig, J.T.: On the consistency of twisted generalized Weyl algebras. Proc. Am. Math. Soc. 140(10), 3349–3363 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Futorny, V., Hartwig, J.T.: Multiparameter twisted Weyl algebras. J. Algebra 357, 69–93 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hartwig, J.T.: Locally finite simple weight modules over twisted generalized Weyl algebras. J. Algebra 303(1), 42–76 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hartwig, J.T.: Twisted generalized Weyl algebras, polynomial Cartan matrices and Serre-type relations. Commun. Algebra 38(12), 4375–4389 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hartwig, J.T., Serganova, V.: Twisted generalized Weyl algebras and primitive quotients of enveloping algebras. arXiv:1504.05361, preprint (2015)
  10. 10.
    Jordan, D.A.: Primitivity in skew Laurent polynomials rings and related rings. Mathematische Zeitschrift 213, 353–371 (1993)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mazorchuk, V., Turowska, L.: Simple weight modules over twisted generalized Weyl algebras. Commun. Algebra 27(6), 2613–2625 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Mazorchuk, V., Turowska, L.: \(\ast \)-Representations of twisted generalized Weyl constructions. Algebras Rep. Theory 5, 163–186 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mazorchuk, V., Ponomarenko, M., Turowska, L.: Some associative algebras associated to \(U(g)\) and twisted generalized Weyl algebras. Mathematica Scandinavica 92, 5–30 (2003)MathSciNetMATHGoogle Scholar
  14. 14.
    Podleś, P.: Quantum spheres. Lett. Math. Phys. 14(3), 193–202 (1987)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Rosenberg, A.L.: Noncommutative Algebraic Geometry and Representations of Quantized Algebras. Kluwer, Dordrecht (1995)CrossRefMATHGoogle Scholar
  16. 16.
    Sergeev, A.: Enveloping algebra \(U(\mathfrak{gl}(3))\) and orthogonal polynomials in several discrete indeterminates. In: Duplij, S., Wess, J. (eds.) Noncommutative Structures in Mathematics and Physics. Proceedings of NATO Advanced Research Workshop, Kiev, 2000. Kluwer, 2001, pp 113–124. arXiv:math/0202182v1 [math.RT]

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of MathematicsUniversity of California RiversideRiversideUSA

Personalised recommendations