Odd-order extensions, monotone-path quantum-torus representations, and scattering-model connections

Investigate the odd-order analogue of the even root-of-unity Kaleidoscope Yang–Baxter algebra, characterize the class of quantum-torus representations that yield monotone-path algebras, and establish a more direct connection between the radical layers of these algebras and the underlying scattering model.

Background

The paper completely analyzes the even-order root-of-unity construction: it proves the sharp chained-word threshold, determines the rank and survival criterion at the sharp layer, identifies the generated algebra as a monotone-path algebra, and computes its radical filtration. The authors emphasize that the parity element and the two endpoint levels are essential to this even-order construction.

The concluding remarks identify three directions that are not resolved by the paper: extending the construction to odd order, determining which quantum-torus representations produce monotone-path algebras, and clarifying how the radical layers reflect the underlying scattering model. These are presented collectively as remaining open problems, rather than as results established in the paper.