Geometry of Higher Dimensional Representations of Noncommutative Algebras

Abstract

In this thesis we begin by looking at polynomials of m n x n matrices invariant under simultaneous conjugation. Thereafter we connect the invariant polynomials to simple representations of noncommutative plane curves, foremost in dimension 1 and 2. We also give an algorithm to find the trace of an arbitrary polynomial in two 2 x 2 matrices. To get a more complete understanding of the simple representations, we consider extensions. In chapter 5 we give a picture of the 2-dimensional simple representations in the case when the algebra is given by k / I, where I is generated by f(x,y) = x^2 + y^2 - 1 + d[x,y] with d in k.