Quantum van Est Isomorphism

TL;DR

This paper extends the classical van Est isomorphism to the realm of quantum groups, establishing isomorphisms between Hopf-cyclic (co)homologies of quantized function algebras and universal enveloping algebras.

Contribution

It introduces a quantum version of the van Est isomorphism linking Hopf-cyclic (co)homologies of quantized algebras in both $h$-adic and $q$-deformation settings.

Findings

Established quantum van Est isomorphisms for $h$-adic deformations.

Extended classical results to $q$-deformation frameworks.

Unified the understanding of (co)homologies in quantum group theory.

Abstract

Motivated by the fact that the Hopf-cyclic (co)homologies of function algebras over Lie groups and universal enveloping algebras over Lie algebras capture the Lie group and Lie algebra (co)homologies, we hereby upgrade the classical van Est isomorphism to ones between the Hopf-cyclic (co)homologies of quantized algebras of functions and quantized universal enveloping algebras, both in $h$-adic and $q$-deformation frameworks.