O ct 2 00 0 QUANTUM DIFFERENTIAL OPERATORS ON THE QUANTUM PLANE

The universal enveloping algebra U (G) of a Lie algebra G acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl 2. Let q be a transcendental element over Q, and let k be a field extension of Q(q) containing √ q. Let U q denote the quantum group corresponding to the Lie algebra sl 2 (k). Let R = kx, yxy−qyx. We will call R the coordinate ring of the quantum plane or sometimes just the quantum plane. This ring R is a representation ring of U q ; that is, every type-1, irreducible, finite dimensional representation of U q appears in R exactly once. Hence, U q acts on the quantum plane. This action is through the quantum-(or q-) differential operators (Section 3.3 of [LR1]). The weight space of U q is Z, the group of integers. Thus R is Γ = Z × Z-graded as deg(x) = (1, 1) and deg(y) = (−1, 1). This corresponds to the fact that x (resp. y) can be seen as the highest (resp. lowest) weight vector of weight 1 (resp.-1) of the unique, type-1, simple, 2-dimensional module. The ring R is also graded by the subgroup Λ generated by (1, 1) and (−1, 1). In this paper we compute the ring (denoted by D Λ q (R)) of q-differential operators of R viewing R as Λ-graded, and study its properties. Note that the ring D Λ q (R) differs from the ring D Γ q (R) of q-differential operators on Γ-graded R. We shall address this in Section 1. In Section 1, we give the requisite preliminaries. Section 2 deals with the description of first order q-differential operators on Γ-graded R. In Section 3 we find the generators of D Λ q (R) explicitly. In Section 4 we describe some basic properties of D Λ q (R) as a ring. In particular, we show that D Λ q (R) is isomorphic to D q (k[t]) k D q (k[t]) as rings (the ring D q (k[t]) has been studied in [IM]) which is simple by Proposition 4.0.1. We …