Quantum K\"ahlerian Lie groups from multiplicative unitaries

We show that the deformation theory of Fr\'echet algebras for actions of K\"ahlerian Lie groups developed by two of us, leads in a natural way to examples of non-compact locally compact quantum groups. This is achieved by constructing a manageable multiplicative unitary out of the Fr\'echet deformation of $C_0(G)$ for the action $\lambda\otimes \rho$ of $G\times G$ and the undeformed coproduct. We also prove that these quantum groups are isomorphic to those constructed out of the unitary dual $2$-cocycle discovered by Neshveyev and Tuset and associated with Bieliavsky's covariant $\star$-product, via the De Commer's results.