Quantum Dilogarithm Identities at Root of Unity

TL;DR

This paper investigates the behavior of quantum dilogarithm identities at roots of unity, establishing new identities for cyclic and non-compact quantum dilogarithms through cluster algebra degenerations.

Contribution

It introduces novel identities for quantum dilogarithms at roots of unity, expanding understanding of their algebraic structures and relations.

Findings

Proved identities for cyclic dilogarithm at roots of unity

Derived new identities for non-compact quantum dilogarithm at b=1

Enhanced understanding of quantum dilogarithm degenerations

Abstract

We study the root of unity degeneration of cluster algebras and quantum dilogarithm identities. We prove identities for the cyclic dilogarithm associated with a mutation sequence of a quiver, and as a consequence new identities for the non-compact quantum dilogarithm at $b=1$.