## Translation Functors and the Shapovalov Determinant

Last updated: 10 February 2015

This is an excerpt from the PhD thesis Translation Functors and the Shapovalov Determinant by Emilie Wiesner, University of Wisconsin-Madison, 2005.

## Contents

 0.1. Abstract 0.2. Introduction

Chapter 1. Basics and Background

 1.1. Some Definitions for Lie Algebras 1.2. Semisimple Lie Algebras 1.3. Odds and Ends

Chapter 2. Lie Algebras with Triangular Decomposition

 2.1. The Category $𝒪$ 2.2. Special Filtrations in Category $𝒪$ 2.3. Blocks 2.4. The Contravariant Form and the Shapovalov Determinant 2.5. Jantzen Filtrations 2.6. Translation Functors

Chapter 3. The Virasoro Algebra

 3.1. Category $𝒪$ 3.2. Affine Lie Algebras 3.3. The Virasoro Algebra and Affine Lie Algebras 3.4. A Determinant Formula for $M\left(\lambda \right)$ 3.5. Blocks 3.6. Translation Functors

Chapter 4. Quantum Groups

 4.1. The Restricted Integral Form ${U}_{𝒜}^{\text{res}}\left(𝔤\right)$ 4.2. A PBW Basis for ${U}_{𝒜}^{\text{res}}\left(𝔤\right)$ 4.3. The Category $𝒪$ 4.4. Embeddings of Verma Modules 4.5. The Shapovalov Determinant 4.6. Translation Functors

Bibliography

## Notes and References

This is an excerpt from the PhD thesis Translation Functors and the Shapovalov Determinant by Emilie Wiesner, University of Wisconsin-Madison, 2005.