Title: Differential operators over quantum spaces

Author: Alexander Verbovetsky

Due to the analogy between quantum case at roots of unity and the classical case in positive characteristic, one should expect the appearance of differential operators that are not compositions of operators of order 1, whereas the dimension of the whole algebra must be the same as in non-deformed case. In this short paper it is shown that this intuition is a true one: we quantize the standard algebraic definition of differential operator by replacing the usual commutators by twisted ones and obtain the algebra of differential operators that has the classical dimension for any $q$ and coincides with the Wess-Zumino algebra at generic parameter values. A detailed description of this algebra is presented. Note also that having such quantum differential operators one can construct quantum jets, de Rham and Spencer complexes, integral forms, Euler operator, and so on, following the same logic as in non-deformed case.

To appear in Acta Applicandae Math.

10 pages. AmS-TeX. To compile source files (.tex) one needs the title page style files titams.tex.