Quantum Kählerian Lie groups from multiplicative unitaries

Abstract

We show that the deformation theory of Fréchet algebras for actions of Kählerian 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 Fréchet deformation of $C_0(G)$ for the action $λ\otimes ρ$ 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.

The present construction was based in an erroneous claim that the dual $2$-cocycle underlying the equivariant quantization of Kählerian Lie groups is unitary. In fact, this cocycle is only a co-isometry. It turns that the deformed fundamental unitary that we have constructed is not unitary as well (but still satisfies the pentagonal equation)