| 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⟨ 1u t, t-1⟩ ⊂ Q, and then move on to show that any Z -graded maximal order A ⊂ 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]. |
