QUANTUM DIFFERENTIAL OPERATORS

UMA N. IYER

Following the definition of quantum differential operators given in [LR], in [IM1], we show that the ring of quantum differential operators on the affine line is the ring generated by x and ∂, the familiar differential operators on the line, along with two additional operators which we call ∂^{β^{1}} and ∂^{β^{−1}}. We describe this ring both as a subring of the ring of graded endomorphisms and as a ring given by generators and relations. From this starting point we are able to describe the ring of quantum differential operators on affine n space and to construct the ring of global quantum differential operators on ℙ^{1}. In [IM2] we investigate quantum differential operators on the skew polynomial ring 𝕜 < x, y > /xy = qyx.

References

IM1. U.Iyer, T. McCune Quantum differential operators on 𝕜[x], To appear in Int. J. Math. IM2. U.Iyer, T. McCune Quantum differential operators on the quantum plane, Preprint. LR. Valery Lunts, Alexander Rosenberg, Differential operators on noncommutative rings, Se-

lecta Math.(N.S) 3, 335–359 (1997).

   E-mail address: uiyer@mpim-bonn.mpg.de

Max-Planck Institut fur Mathematik, Vivatsgasse 7, 53111 Bonn.

Date: February 18, 2002.