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Stable Anti-Yetter-Drinfel'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), 587-590 Abstract We define and study a class of entwined modules (stable anti-Yetter-Drinfel'd modules) that serve as coefficients for the Hopf-cyclic homology and cohomology. In particular, we explain their relationship with Yetter-Drinfel'd moudles and Drinfel'd doubles. Among sources of examples of stable anti-Yetter-Drinfel'd modules, we find Hopf-Galois extensions with a flipped version of the Miyashita-Ulbrich action.

	Introduction The aim of this paper is to define and provide sources of examples of stable anti-Yetter-Drinfel'd modules. They play the role of coefficients for Hopf-cyclic theory [7]. In particular, we claim that modular pairs in involution of Connes and Moscovici are precisely 1-dimensional stable anti-Yetter-Drinfel'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 _MΔ(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.