Quantum van Est isomorphism

Motivated by the fact that the Hopf-cyclic (co)homology of the quantized algebras of functions and quantized universal enveloping algebras are the correct analogues of the Lie algebra and Lie group (co)homologies, we hereby construct three van Est type isomorphisms between the Hopf-cyclic (co)homologies of Lie groups and Lie algebras, and their quantum groups and corresponding enveloping algebras, both in h-adic and q-deformation frameworks.