UC San Diego

The first Weyl Algebra can be viewed to have Z-graded quotient ring Q=k(u)[t, t⁻¹; sigma], and Bell and Rogalski have classified all simple Z-graded subrings of this quotient ring with Gelfand-Kirillov (GK) dimension 2. In this paper, we seek to understand maximal orders of this quotient ring with GK dimension 3. We start by examining a representative example, k viewed as a subset of Q, and then move on to show that any Z-graded maximal order A in Q must have A₀ be a localization of k[u], or a ring in the form k[S], where S is a sigma- closed set of rational functions of the form 1/(u-a). Finally, we completely classify the possible Z-graded maximal orders inside k(u)[t, t⁻¹; sigma]