Braiding Operator via Quantum Cluster Algebra

TL;DR

This paper constructs a braiding operator using quantum cluster algebra and quantum dilogarithm, demonstrating its relation to hyperbolic geometry and its reduction to Kashaev R-matrix at roots of unity.

Contribution

It introduces a new braiding operator based on quantum cluster algebra that generalizes known R-matrices and connects quantum algebra with hyperbolic geometry.

Findings

The braiding operator is a q-deformation of the R-operator.

At roots of unity, it reduces to the Kashaev R-matrix.

Establishes a link between quantum cluster algebra and hyperbolic geometry.

Abstract

We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octrahedron is assigned. Also shown is that, by taking q to be a root of unity, our braiding operator reduces to the Kashaev R-matrix up to a simple gauge-transformation.