Abstract
In this paper, we show that total integrals and cointegrals are new sources of stable anti-Yetter–Drinfeld modules. We explicitly show that how special types of total (co)integrals can be used to provide both (stable) anti Yetter–Drinfeld and Yetter–Drinfeld modules. We use these modules to classify total (co)integrals and (cleft) Hopf Galois (co)extensions of the Connes–Moscovici Hopf algebra, and some examples of universal enveloping algebra and polynomial algebra.
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