Classification of Line Modules and Finite Dimensional Simple Modules over a Deformation of the Polynomial Ring in Three Variables

Abstract

Let $\Bbbk$ be a field and $A$ the non-commutative $\Bbbk$-algebra generated by $x_1, x_2, x_3$ subject to the relations $$ q x_ix_j - q^{-1} x_jx_i \; = \; x_k $$ as $(i,j,k)$ ranges over all cyclic permutations of $(1,2,3)$, where $q\in \Bbbk - \{ 0\}$. This thesis sets out to understand the representation theory of $A$. In particular, we classify all finite dimensional simple modules over $A$ when $q$ is not a root of unity. To this end, we introduce the notion of a linear module over a filtered $\Bbbk$-algebra, an analogue to the notion of a linear module defined for a connected graded $\Bbbk$-algebra. Finite dimensional simple $A$-modules are closely related to certain linear modules for $A$ of Gelfand-Kirillov dimension one, which we call line modules, in the sense that every finite dimensional simple $A$-module $V$ appears in an exact sequence $$ 0 \;\longrightarrow\; M' \;\longrightarrow \; M \; \longrightarrow \; V \; \longrightarrow \; 0, $$ in which $M$ and $M'$ are line modules for $A$. The main result shows that there are five non-isomorphic simple $A$-modules of each dimension.