$\varrho$-Poincar\'e: bicrossproduct structure, $\star$-products and quantum Lie algebra

TL;DR

This paper explores the bicrossproduct structure and noncommutative $ ext{ extsterling}$-products of the $ ho$-Poincaré quantum group, extending to general Lie algebra deformations and analyzing their algebraic and geometric features.

Contribution

It introduces a comprehensive analysis of the bicrossproduct structure, $ ext{ extsterling}$-products, and quantum Lie algebra for the $ ho$-Poincaré quantum group, extending previous models.

Findings

Relation between different bases of the quantum universal enveloping algebra

Analysis of noncommutative $ ext{ extsterling}$-products on $ ho$-Minkowski spacetime

Introduction of the $ ho$-Poincaré quantum Lie algebra

Abstract

We discuss the bicrossproduct structure of the quantum group $\varrho$-Poincar\'e and of the dual quantum universal enveloping algebra, expanding the construction to general Lie algebra-type deformations of Poincar\'e coming from classical $r$-matrices. We review the relation between different bases of the quantum universal enveloping algebra of $\varrho$-Poincar\'e and noncommutative $\star$-products defined on the $\varrho$-Minkowski spacetime, analysing some of their relevant features. Furthermore, we comment on the role of physical bases and introduce the $\varrho$-Poincar\'e quantum Lie algebra.