nLab bicovariant differential calculus

Definitions

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

• Stanisław Lech Woronowicz: Differential calculus on compact matrix pseudogroups (quantum groups), Commun. Math. Phys. 122 (1989) 125–170 [euclid:cmp/1104178320, MR0994499]

• Shahn Majid: Classification of bicovariant differential calculi, Journal of Geometry and Physics 25 1–2 (1998) 119–140 [doi q-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

• Ursula Carow-Watamura, Satoshi Watamura, Complex quantum group, dual algebra and bicovariant differential calculus, Comm. Math. Phys. 151:3 (1993) 487–514 euclid

• Peter Schauenburg, Hopf modules and Yetter–Drinfel′d modules, J. Algebra 169:3 (1994) 874–890 doi

In monoidal additive categories:

• Keegan J. Flood, Gabriele Lobbia, Giacomo Tendas: Canonical differential calculi via functorial geometrization [arXiv:2512.20742]

category: algebra