# Cylindrical Dyck paths and the Mazorchuk–Turowska equation

@article{Hartwig2015CylindricalDP,
title={Cylindrical Dyck paths and the Mazorchuk–Turowska equation},
author={Jonas Torbj{\"o}rn Hartwig and Daniele Rosso},
journal={Journal of Algebraic Combinatorics},
year={2015},
volume={44},
pages={223-247}
}
• Published 15 July 2015
• Mathematics
• Journal of Algebraic Combinatorics
We classify all solutions (p, q) to the equation $$p(u)q(u)=p(u+\beta )q(u+\alpha )$$p(u)q(u)=p(u+β)q(u+α) where p and q are complex polynomials in one indeterminate u, and $$\alpha$$α and $$\beta$$β 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…
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## References

SHOWING 1-10 OF 18 REFERENCES
Counting generalized Dyck paths
The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from $(0,0)$ to $(n,n)$ which is below the diagonal line $y=x$. One way to
Twisted Generalized Weyl Algebras and Primitive Quotients of Enveloping Algebras
• Mathematics
• 2015
To each multiquiver Γ we attach a solution to the consistency equations associated to twisted generalized Weyl (TGW) algebras. This generalizes several previously obtained solutions in the
On the consistency of twisted generalized Weyl algebras
• Mathematics
• 2011
A twisted generalized Weyl algebra A of degree n depends on a base algebra R, n commuting automorphisms s_i of R, n central elements t_i of R and on some additional scalar parameters. In a paper by
Enveloping algebra U(gl(3)) and orthogonal polynomials in several discrete indeterminates
Let A be an associative complex algebra and L an invariant linear functional on it (trace). Let i be an involutive antiautomorphism of A such that L(i(a))=L(a) for any a in A. Then A admits a
Enveloping Algebra of GL(3) and Orthogonal Polynomials
Let A be an associative algebra over C and L an invariant linear functional on it (trace). Let ω be an involutive antiautomorphism of A such that L(ω(a)) = L(a) for any a ω A. Then A admits a
Twisted Generalized Weyl Algebras, Polynomial Cartan Matrices and Serre-Type Relations
A twisted generalized Weyl algebra (TGWA) is defined as the quotient of a certain graded algebra by the maximal graded ideal I with trivial zero component, analogous to how Kac–Moody algebras can be
Some associative algebras related to $U(\mathfrak g)$ and twisted generalized Weyl algebras
• Mathematics
• 2003
We prove that both Mickelsson step algebras and Orthogonal Gelfand-Zetlin algebras are twisted generalized Weyl algebras. Using an analogue of the Shapovalov form we construct all weight simple
Noncommutative Algebraic Geometry and Representations of Quantized Algebras
Preface. I. Noncommutative Affine Schemes. II. The Left Spectrum and Irreducible Representation of `Small' Quantized and Classical Rings. III. Noncommutative Local Algebra. IV. Noncommutative Local