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Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

Samuel A. Lopes

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

Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants

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 .
Paper Structure (47 sections, 86 equations, 2 figures, 1 table)

This paper contains 47 sections, 86 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. The Weyl algebra: definition
  4. The Weyl algebra: representations
  5. char(F)=0
  6. char(F)=p>0
  7. The Weyl algebra: automorphisms and derivations
  8. The Dixmier and Jacobian conjectures
  9. The Weyl algebra: combinatorics
  10. The Weyl algebra: ring theoretical properties
  11. Deformation theory and Hochschild (co)homology
  12. Representation Theory
  13. Quantized enveloping algebras
  14. Primitive ideals of Uq(sln+)
  15. Infinite-dimensional representations of Uq(sln+)
  16. ...and 32 more sections

Figures (2)

  • Figure 1: Young's lattice
  • Figure 2: The lattice of inclusions in the family $\mathsf{A}_h$