Heisenberg Double ℋ(B*) as a Braided Commutative Yetter–Drinfeld Module Algebra Over the Drinfeld Double

Abstract

We study the Yetter–Drinfeld 𝒟(B)-module algebra structure on the Heisenberg double ℋ(B*) endowed with a “heterotic” action of the Drinfeld double 𝒟(B). In terms of the braiding of Yetter–Drinfeld modules, ℋ(B*) is braided commutative. By the Brzeziński–Militaru theorem, ℋ(B*) # 𝒟(B) is then a Hopf algebroid over ℋ(B*). The heterotic action can also be interpreted in the spirit of Lu's description of ℋ(B*) as a twist of 𝒟(B). 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 . In particular, it follows that Mat p (ℂ) is a braided commutative Yetter–Drinfeld -module algebra and is a Hopf algebroid over Mat p (ℂ).