This is an abridged version of our Habilitation thesis. In these notes, we aim to summarize our research interests and achievements as well as motivate what drives our work: symmetry, structure and invariants. The paradigmatic example which permeates and often inspires our research is the Weyl algebra $\mathbb{A}_{1}$.

• 16S32 - Rings of differential operators (associative algebraic aspects)
• 16S36 - Ordinary and skew polynomial rings and semigroup rings
• 17B37 - Quantum groups (quantized enveloping algebras) and related deformations