Heisenberg double H(B^*) as a braided commutative Yetter-Drinfeld module algebra over the Drinfeld double

Details Heisenberg double H(B^*) as a braided commutative Yetter-Drinfeld module algebra over the Drinfeld double Heisenberg double H(B^*) as a braided commutative Yetter-Drinfeld module algebra over the Drinfeld double

2010-01-03

arxiv.org/abs/1001.0733v2

Communications in Algebra, Volume 39 (2011) Number 5, pp. 1883-1906

Mathematics

Quantum Algebra

22 pages, amsart++, times. V2: minor changes; reference to an arxiv number corrected

Scientific paper

10.1080/00927872.2010.480955

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). This action can be interpreted in the spirit of Lu's description of H(B^*) as a twist of D(B). In terms of the braiding of Yetter--Drinfeld modules, H(B^*) is braided commutative. By the Brzezinski--Militaru theorem, H(B^*)#D(B) is then a Hopf algebroid over H(B^*). 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_q(sl(2)). In particular, it follows that Mat_p(C) is a braided commutative Yetter--Drinfeld U_q(sl(2))-module algebra and Mat_p(U_q(sl(2))) is a Hopf algebroid over Mat_p(C).

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