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Stable AntiYetterDrinfel'd Modules
Piotr Hajac Masoud Khalkhali
Bahram Rangipour Yorck Sommerhäuser


 Preprint: Institute of Mathematics of the Polish Academy of Sciences:
IM PAN 647
 Preprint: XXX preprint archive:
math.QA/0405005
 Journal: C. R. Acad. Sci., Paris, Sér I, Math. 338 (2004), 587590
Abstract
We define and study a class of entwined modules (stable antiYetterDrinfel'd modules) that serve as coefficients for the Hopfcyclic homology and cohomology. In particular, we explain their relationship with YetterDrinfel'd moudles and Drinfel'd doubles. Among sources of examples of stable antiYetterDrinfel'd modules, we find HopfGalois extensions with a flipped version of the MiyashitaUlbrich action.




Introduction
The aim of this paper is to define and provide sources of examples of stable antiYetterDrinfel'd modules. They play the role of coefficients for Hopfcyclic theory [7]. In particular, we claim that modular pairs in involution of Connes and Moscovici are precisely 1dimensional stable antiYetterDrinfel'd modules.
Throughout the paper we assume that H is a Hopf algebra with a bijective antipode. On the one hand, the bijectivity of the antipode is implied by the existence of a modular pair in involution, so that then it need not be assumed. On the other hand, some parts of arguments might work even if the antipode is not bijective. We avoid such discussions. The coproduct, counit and antipode of H are denoted by Δ, ε and S, respectively. For the coproduct we use the notation
Δ(h) =h^{(1)} ⊗ h^{(2)},
for a left coaction on M we write
_{M}Δ(m) = m^{(1)} ⊗ m^{(0)},
and for a right coaction
Δ_{M}(m) = m^{(0)} ⊗ m^{(1)}.
The summation symbol is suppressed everywhere. We assume all algebras to be associative, unital and over the same ground field k. The symbol O(X) stands for the algebra of polynomial functions on X.



