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Motivated by the fact that the Hopf-cyclic (co)homology of the quantizedalgebras of functions and quantized universal enveloping algebras are thecorrect analogues of the Lie algebra and Lie group (co)homologies, we herebyconstruct three van Est type isomorphisms between the Hopf-cyclic(co)homologies of Lie groups and Lie algebras, and their quantum groups andcorresponding enveloping algebras, both in $h$-adic and $q$-deformationframeworks.

Quantum van Est Isomorphism