Star-products for Lie-algebraic noncommutative Minkowski space-times

TL;DR

This work develops a Weyl-quantization, convolution-algebra approach to construct star-products and involutions for Lie-algebraic noncommutative Minkowski space-times associated with 11 of the 17 classes of coordinate algebras. A central result is that unimodular coordinate groups admit the ordinary Lebesgue trace, while nonunimodular groups require a KMS-weight twisted trace, reflecting a modular time evolution in the latter case. The authors provide explicit star-product formulas and involutions across unimodular and nonunimodular spaces, and derive Poincaré Hopf algebras compatible with the deformed algebras in several cases, clarifying the symmetry structure and potential field-theory applications on these quantum space-times. They also discuss the relation to Drinfeld twists and the limitations of closure for certain spaces, outlining future work on the remaining models and their physical implications. $\$

Abstract

Poisson structures of the Poincaré group can be linked to deformations of the Minkowski space-time, classified some time ago by Zakrewski. Based on this classification, various quantum Minkowski space-times with coordinates Lie algebras and specific Poincare Hopf algebras have been exhibited by Mercati and called T-Minkowski space-times. Here we construct the star products and involutions characterizing the $\star$-algebras for a broad family of Lie algebras which includes 11 out of 17 Lie algebras of T-Minkowski spaces. We show that the usual Lebesgue integral defines either a trace or a KMS weight ('twisted trace') depending on whether the Lie group of the coordinates' Lie algebra is unimodular or not. Finally, we give the Poincaré Hopf algebras when they are compatible with our $*$-product. General derivation of such symmetry Hopf algebras are briefly discussed.