A Dixmier-Moeglin equivalence for skew Laurent polynomial rings

The work of Dixmier in 1977 and Moeglin in 1980 show us that for a prime ideal $P$ in the universal enveloping algebra of a complex finite-dimensional Lie algebra the properties of being primitive, rational and locally closed in the Zariski topology are all equivalent. This equivalence is referred to as the Dixmier-Moeglin equivalence. In this thesis we will study skew Laurent polynomial rings of the form ${\mathbb{C}}[x_1,\ldots,x_d][z,z^{-1};\sigma]$ where $\sigma$ is a ${\mathbb{C}}$-algebra automorphism of ${\mathbb{C}}[x_1,\ldots,x_d]$. In the case that $\sigma$ restricts to a linear automorphism of the vector space ${\mathbb{C}} + {\mathbb{C}}x_1 + \cdots + {\mathbb{C}}x_d$, we show that the Dixmier-Moeglin equivalence holds for the prime ideal (0).