Planar algebras, cabling and the Drinfeld double

TL;DR

This paper constructs an explicit embedding of the planar algebra associated with the Drinfeld double of a finite-dimensional Hopf algebra into a two-cabling of the dual Hopf algebra's planar algebra, and characterizes its image.

Contribution

It provides a new explicit embedding and characterization of the planar algebra of the Drinfeld double in terms of the dual Hopf algebra's planar algebra.

Findings

Explicit embedding of Drinfeld double planar algebra into two-cabling of dual Hopf algebra's planar algebra

Characterization of the image of this embedding

Enhanced understanding of the structure of Drinfeld doubles in planar algebra terms

Abstract

We produce an explicit embedding of the planar algebra of the Drinfeld double of a finite-dimensional, semisimple and cosemisimple Hopf algebra $H$ into the two-cabling of the planar algebra of the dual Hopf algebra $H^*$ and characterise the image.