The Ribbon Elements of Drinfeld Double of Radford Hopf Algebra

TL;DR

This paper characterizes when the Drinfeld double of Radford's Hopf algebra has ribbon elements, showing it depends on the parity of the integer parameters, and explicitly computes these ribbon elements.

Contribution

It provides a complete characterization of the existence and number of ribbon elements in the Drinfeld double of Radford Hopf algebras, including explicit computations.

Findings

Ribbon elements exist if and only if n is odd.

Number of ribbon elements depends on the parity of m and n.

Explicit formulas for all ribbon elements are provided.

Abstract

Let $m$, $n$ be two positive integers, $\Bbbk$ be an algebraically closed field with char($\Bbbk)\nmid mn$. Radford constructed an $mn^{2}$-dimensional Hopf algebra $R_{mn}(q)$ such that its Jacobson radical is not a Hopf ideal. We show that the Drinfeld double $D(R_{mn}(q))$ of Radford Hopf algebra $R_{mn}(q)$ has ribbon elements if and only if $n$ is odd. Moreover, if $m$ is even and $n$ is odd, then $D(R_{mn}(q))$ has two ribbon elements, if both $m$ and $n$ are odd, then $D(R_{mn}(q))$ has only one ribbon element. Finally, we compute explicitly all ribbon elements of $D(R_{mn}(q))$.