Quantum K\"ahlerian Lie groups from multiplicative unitaries

TL;DR

This paper develops a deformation approach for Kählerian Lie groups that naturally produces non-compact quantum groups, connecting multiplicative unitaries, dual cocycles, and covariant star-products.

Contribution

It introduces a method to construct non-compact quantum groups from Kählerian Lie groups using Fréchet algebra deformations and multiplicative unitaries, linking existing quantum group frameworks.

Findings

Constructed manageable multiplicative unitaries from deformed $C_0(G)$

Proved quantum groups are isomorphic to those from dual 2-cocycles

Connected deformation approach to covariant star-products

Abstract

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.