Equivariant Hopf Galois extensions and Hopf cyclic cohomology

We define the notion of equivariant Hopf Galois extension and apply it as a functor between category of SAYD modules of the Hopf algebras involving in the extension. This generalizes the result of Jara-Stefan and B"ohm-Stefan on associating a SAYD modules to any ordinary Hopf Galois extension.

💡 Research Summary

The paper introduces a new concept called an “equivariant Hopf‑Galois extension” and shows how it yields a functorial correspondence between the categories of stable anti‑Yetter‑Drinfeld (SAYD) modules attached to the two Hopf algebras that appear in the extension. The work is motivated by Hopf cyclic cohomology, a non‑commutative analogue of Connes–Moscovici’s cyclic cohomology, where SAYD modules serve as coefficient objects. Earlier results by Jara‑Stefan and Böhm‑Stefan demonstrated that an ordinary Hopf‑Galois extension gives rise to a functor from the SAYD‑category of the acting Hopf algebra to the SAYD‑category of the base algebra’s Hopf symmetry. However, those constructions treat only a single Hopf algebra at a time.

The authors begin by recalling the necessary background: Hopf algebras, module algebras, comodule algebras, and the precise definition of a SAYD module (the stability condition and the anti‑Yetter‑Drinfeld condition). They then define an equivariant Hopf‑Galois extension (B\subseteq A) relative to a pair of Hopf algebras ((H,K)). The data consist of an (H)-comodule algebra structure on (A) and a (K)-module algebra structure on (B) together with a compatibility (equivariance) condition that forces the two actions to commute. The usual Galois map (\beta: A\otimes_B A\to A\otimes H) is required to be bijective, and the subalgebra of (H)-coinvariants is precisely (B). This definition reduces to the classical Hopf‑Galois extension when (K) is the trivial Hopf algebra (the ground field) and to a self‑equivariant situation when (H=K).

The central theorem (Theorem 4.1) states that for any (H)-SAYD module (M) there is a canonical construction
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