On quantized algebra of Wess-Zumino differential operators at roots of unity (Q1383896)

This paper is devoted to the study of differential operators on quantum hyperplanes. The author first recalls the definition of a quantum affine space \(A\) via \(R\)-matrices as well as the construction of a quantum group \(H\) acting on \(A\). Next, he gives an algebraic definition of differential operators of order \(\leq k\) and shows that the spaces of such differential operators can be canonically endowed with the structure of a left \(H\)-module and of an \(A\)-bimodule. Next, he also makes the spaces of differential operators into a right \(H\)-comodule and studies basic properties of differential operators. Finally, the quantum Weyl algebra is introduced and shown to be isomorphic to the algebra of all differential operators if the deformation parameter \(q\) satisfies either that \(q^2\) is not a root of unity or \(q^2=1\) in characteristic zero. If \(q^2\) is a primitive root of unity the kernel of the canonical map from the Weyl algebra to the algebra of differential operators is explicitly described. The results are compared to those of \textit{J. Wess, B. Zumino} [Covariant differential calculus on the quantum hyperplane, Nucl. Phys. B Proc. Suppl. 18, 302-312 (1990)].