$\varrho$-Poincaré: bicrossproduct structure, $\star$-products and quantum Lie algebra

We discuss the bicrossproduct structure of the quantum group $\varrho$-Poincaré and of the dual quantum universal enveloping algebra, expanding the construction to general Lie algebra-type deformations of Poincaré coming from classical $r$-matrices. We review the relation between different bases of the quantum universal enveloping algebra of $\varrho$-Poincaré 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 $\varrho$-Poincaré quantum Lie algebra.