Quantum generalized cluster algebras and quantum dilogarithms of higher degrees

TL;DR

This paper extends the quantization framework of cluster algebras to generalized versions and introduces higher-degree quantum dilogarithms, establishing identities linked to quantum Y-seed periods.

Contribution

It introduces quantum dilogarithms of higher degrees and integrates them into the quantization of generalized cluster algebras, expanding the theoretical framework.

Findings

Derived identities of generalized quantum dilogarithms

Connected quantum dilogarithm identities with quantum Y-seed periods

Extended quantization to Chekhov-Shapiro generalized cluster algebras

Abstract

We extend the notion of the quantization of the coefficients of the ordinary cluster algebras to the generalized cluster algebras by Chekhov and Shapiro. In parallel to the ordinary case, it is tightly integrated with certain generalizations of the ordinary quantum dilogarithm, which we call the quantum dilogarithms of higher degrees. As an application, we derive the identities of these generalized quantum dilogarithms associated with any period of quantum Y-seeds.