ℤ-graded simple rings

Abstract

The Weyl algebra over a field k k of characteristic 0 0 is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all Z \mathbb {Z} -graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study Z \mathbb {Z} -graded simple rings A A of any dimension which have a graded quotient ring of the form K [ t , t − 1 ; σ ] K[t, t^{-1}; \sigma ] for a field K K . Under some further hypotheses, we classify all such A A in terms of a new construction of simple rings which we introduce in this paper. In the important special case that GKdim ⁡ A = t r . d e g ⁡ ( K / k ) + 1 \operatorname {GKdim} A = \operatorname {tr.deg}(K/k) + 1 , we show that K K and σ \sigma must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry.