Let be a -algebra. A first order differential calculus is an -bimodule with an -derivation such that is -spanned by elements of the form where .
Let be a Hopf algebra. A first order differential calculus is bicovariant if is also an -bicomodule (compatibly left and right comodule) where the left and right coaction , are -bimodule maps and is an -bicomodule map. In other words, is a Hopf bimodule and is a Hopf bimodule map.
It is well-known that the category of Hopf bimodules is equivalent (even as a monoidal category!) to the category of Yetter–Drinfeld modules, at least for finite-dimensional Hopf algebra .
The notion of bicovariance of a noncommutative differential calculus is introduced in
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