The Quantum Cartan Algebra Associated to a Bicovariant Differential Calculus

We associate to any (suitable) bicovariant differential calculus on a quantum group a Cartan Hopf algebra. We provide it with a left, respectively right, representation in terms of left, respectively right, Cartan calculus operators acting on the corresponding exterior algebra. The example of the Hopf algebra associated to the 4D+ differential calculus on SUq(2) is described.