Let $A$ be a $k$-algebra. A first order differential calculus $\Omega^1(A)$ is an $A$-bimodule with an $A$-derivation $d: A\to\Omega^1(A)$ such that $\Omega^1(A)$ is $k$-spanned by elements of the form $a d(b)$ where $a,b\in A$.

Let $A = H$ be a Hopf algebra. A first order differential calculus $d:H \to \Gamma = \Omega^1(H)$ is bicovariant if $\Omega^1(H)$ is also an $H$-bicomodule (compatibly left and right comodule) where the left and right coaction $\rho^L$, $\rho^R$ are $A$-bimodule maps and $d$ is an $A$-bicomodule map. In other words, $\Gamma$ is a Hopf bimodule and $d:A\to \Gamma$ is a Hopf bimodule map.

Relation to Yetter–Drinfeld condition

It is well-known that the category of Hopf bimodules ${}^H_H\mathcal{M}^H_H$ is equivalent (even as a monoidal category!) to the category of Yetter–Drinfeld modules, at least for finite-dimensional Hopf algebra $H$.

Literature

The notion of bicovariance of a noncommutative differential calculus is introduced in

Shahn Majid, Classification of bicovariant differential calculi, Journal of Geometry and Physics 25:1–2, April 1998, 119–140; doiq-alg/9608016

Konrad Schmüdgen, Axel Schüler, Classification of bicovariant differential calculi on quantum groups of type A, B, C and D, Commun. Math. Phys. 167:3, pp 635–670 (1995) euclid