Yetter--Drinfeld structures on Heisenberg doubles and chains

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).