Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

TL;DR

This paper summarizes research on noncommutative algebra and representation theory, focusing on symmetry, structure, and invariants, with the Weyl algebra as a central example, highlighting key achievements and motivations.

Contribution

It provides an overview of advances in understanding the Weyl algebra and related structures within noncommutative algebra and representation theory.

Findings

Insights into the structure of the Weyl algebra

Development of invariants in noncommutative settings

Connections between symmetry and algebraic properties

Abstract

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 $\mathbb{A}_{1}$.