The $\kappa$-Poincar\'e Group on a $C^*$-level

TL;DR

This paper constructs the $C^*$-algebraic $ppa$-Poincare9 Group using groupoid algebras, revealing a twisted comultiplication and providing generators and relations, advancing the algebraic understanding of quantum groups.

Contribution

It introduces a $C^*$-algebraic construction of the $ppa$-Poincare9 Group via differential groupoids, highlighting a twisted comultiplication and explicit generators.

Findings

The $C^*$-algebra matches that of the $ppa$-Euclidean Group.

The comultiplication is twisted by a unitary multiplier.

Generators and their commutation relations are explicitly given.

Abstract

The $C^*$-algebraic $\kappa$-Poincar\'{e} Group is constructed. The construction uses groupoid algebras of differential groupoids associated to Lie group decomposition. It turns out the underlying $C^*$-algebra is the same as for "$\kappa$-Euclidean Group" but a comultiplication is twisted by some unitary multiplier. Generators and commutation relations among them are presented.