arXiv subjects
Publications and source records attributed to Timothy C. McCune.
Following the definitions of the algebras of differential operators, $β$-differential operators, and the quantum differential operators on a noncommutative (graded) algebra given in \cite{LR}, we describe these operators on the free associative algebra. We further study their properties.
Withdrawn on account of a crucial error in the main theorem
Following the definition of quantum differential operators given by Lunts and Rosenberg in (Sel. math., New ser. 3 (1997) 335--359), we show that the ring of quantum differential operators on the affine line is the ring generated by x and \del, the familiar differential operators on the line, along with two additional operators which we call \del^β^1 and \del^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 the projective line.
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.