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

Yetter--Drinfeld structures on Heisenberg doubles and chains