Representations of Quantum Nilpotent Algebras at Roots of Unity, and Their Completely Prime Quotients

This thesis studies algebras contained in a large class of iterated Ore extensions, as well as their quotient algebras by completely prime ideals, and develops methods for computing their \emph{polynomial identity (PI)} degree and constructing irreducible representations of maximal dimension. This class contains quantum nilpotent algebras, including many examples of quantised coordinate rings and quantised enveloping algebras. When the deformation parameters are allowed to be roots of unity these algebras often become PI algebras. We focus our attention on such algebras in this work. By extending Cauchon's deleting derivations algorithm in the generic setting we are able, given a suitable PI algebra $A$ and completely prime ideal $P\lhd A$, to construct a quantum affine space $A'$ and completely prime ideal $Q\lhd A'$, such that the quotient algebras $A/P$ and $A'/Q$ share the same PI degree. This extends a result of Haynal, where existence of $Q$ was proved but no method of construction was provided. The PI degree of several small examples are then calculated. For completely prime quotients of quantum matrices the PI degree is shown to be closely related to properties of \emph{Cauchon-Le diagrams}. We prove that given any Cauchon-Le diagram, the invariant factors of its associated matrix are all powers of $2$. Furthermore, we compute the \emph{toric permutation} of Cauchon-Le diagrams corresponding to \emph{quantum determinantal rings}, which then allows us to state an explicit formula for the PI degree of a quantum determinantal ring at a root of unity. Finally, we show how certain irreducible representations of the quotient $A'/Q$ may be passed through the deleting derivations algorithm to give an irreducible representation of $A/P$, and we construct an irreducible representation of a general quantum determinantal ring.