Cylindrical Dyck paths and the Mazorchuk-Turowska equation

Abstract

We classify all solutions (p,q) to the equation p(u)q(u)=p(u+b)q(u+a) where p and q are complex polynomials in one indeterminate u, and a and b 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.

21 pages

References in corpus (3)

• Locally finite simple weight modules over twisted generalized Weyl algebras
• Twisted generalized Weyl algebras and primitive quotients of enveloping algebras
• Counting generalized Dyck paths

Cited by in corpus (5)

• Noncommutative fiber products and lattice models
• Fixed rings of twisted generalized Weyl algebras
• Classification of twisted generalized Weyl algebras over polynomial rings
• Unitarizability of weight modules over noncommutative Kleinian fiber products
• Sub-bimodules of the identity bimodule for cyclic quivers