  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).
