Yetter--Drinfeld structures on Heisenberg doubles and chains

Abstract

For a Hopf algebra B with bijective antipode, we show that the Heisenberg double HÔB ¦ Õ is a braided commutative Yetter-Drinfeld module algebra over the Drinfeld double DÔBÕ. The braiding structure allows generalizing HÔB ¦ Õ B ¦cop ³ B to "Heisenberg n-tuples" and "chains" . . . ³B ¦cop ³B³B ¦cop ³B³..., all of which are Yetter-Drinfeld DÔBÕ-module algebras. For B a particular Taft Hopf algebra at a 2pth root of unity, the construction is adapted to yield Yetter-Drinfeld module algebras over the 2p 3 -dimensional quantum group U q sℓÔ2Õ.