# Yetter--Drinfeld structures on Heisenberg doubles and chains

## TL;DR

This paper explores the structure of Heisenberg doubles as Yetter--Drinfeld module algebras over the Drinfeld double, generalizing to chains and applying to specific quantum groups at roots of unity.

## Contribution

It introduces a braided commutative structure on Heisenberg doubles and extends this to chains, with applications to quantum groups at roots of unity.

## Key findings

- Heisenberg double H(B^*) is a braided commutative Yetter--Drinfeld module algebra.
- Generalization to Heisenberg n-tuples and chains as Yetter--Drinfeld D(B)-module algebras.
- Application to Taft Hopf algebra and quantum group U_q(sl(2)) at roots of unity.

## Abstract

For a Hopf algebra B with bijective antipode, we show that the Heisenberg double H(B^*) is a braided commutative Yetter--Drinfeld module algebra over the Drinfeld double D(B). The braiding structure allows generalizing H(B^*) = B^{*cop}\braid B to "Heisenberg n-tuples" and "chains" ...\braid B^{*cop}\braid B \braid B^{*cop}\braid B\braid..., all of which are Yetter--Drinfeld D(B)-module algebras. For B a particular Taft Hopf algebra at a 2p-th root of unity, the construction is adapted to yield Yetter--Drinfeld module algebras over the 2p^3-dimensional quantum group U_qsl(2).