# Representations of the Witt-Jacobson Lie-Algebras

By Khalid Rian
PhD Dissertations
September 2011
Abstract:
In this thesis, we study the representations of the Witt-Jacobson Lie algebras $W^n$ of rank $n$. The first part deals with $W^1$ where the main goal is to obtain a classification of the extensions of the simple $U_\chi(W^1)$--modules having character $\chi$ of height at most 1. Here $U_\chi(W^n)$ denotes the reduced enveloping algebra of $W^n$ corresponding to $\chi$. The second part deals with the projective indecomposable modules of $U_\chi(W^n)$ where $n > 1$ and $\chi$ is a character of height 0. The main goal here is to determine the Cartan invariants of $U_\chi(W^n)$. We keep the setting as general as possible, but some of the results are only presented for $n=2$.