Implications of the Hopf algebra properties of noncommutative   differential calculi
  We define a noncommutative algebra of four basic objects within a differential calculus on quantum groups: functions, 1-forms, Lie derivatives and inner derivations, as the cross-product algebra associated with Woronowicz's (differential) algebra of functions and forms. This definition properly takes into account the Hopf algebra structure of the Woronowicz calculus. It also provides a direct proof of the Cartan identity. 
A.A. Vladimirov (JINR, Dubna)