bicovariant differential calculus

Bicovariant differential calculus


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

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


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

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

  • Shahn Majid, Classification of bicovariant differential calculi, Journal of Geometry and Physics 25:1–2, April 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

category: algebra

Last revised on October 14, 2014 at 19:28:10. See the history of this page for a list of all contributions to it.