Differential operators over quantum spaces
The aim of this paper is to construct a differential calculus on certain noncommutative spaces. The starting point of the construction is an associative unital algebra \(A\) which is a comodule algebra over a Hopf algebra \(H\). This structure gives rise to a natural transposition operator \(\sigma_X\colon X\otimes A\to A\otimes X\) for any \(H\)-module \(X\), which is the central ingredient in the theory. It is assumed that \(A\) is braided commutative with respect to the transposition \(\sigma_A\). Moreover, one needs a fixed commuting, invertible, group-like element \(\phi\in H\), which enters in the definition of the transposition of graded objects. Using these ingredients, the author first defines an algebraic notion of a differential operator between two \(A\)-modules, and shows that the space of all differential operators of some fixed order is a left and right module over \(A\). Next, an analog of the Spencer complex is constructed and modules of multiderivations are introduced and studied. Then the author defines modules of jets and modules of \(k\)-forms as representing objects and studies their properties, and introduces an analog of the exterior derivative. To study adjoints of differential operators, the author next introduces the notion of right differential operators. He introduces a module of volume forms and a notion of cohomological integration. Finally, under some additional assumptions a version of Green's formula and of Poincaré duality is proved. An appendix collects some background material on Hopf-algebras. The concepts introduced in the paper are worked out in several examples and in particular for quantum hyperplanes, where the resulting differential operators are shown to be isomorphic to the differential operators introduced by \textit{J. Wess, B. Zumino} [Covariant differential calculus on the quantum hyperplane, Nucl. Phys. B Proc. Suppl. 18, 302-312 (1990)] for generic \(q\). A more detailed elaboration of the case of quantum hyperplanes can be found in the author's paper [ibid. 363-370 (1997; see the following review Zbl 0916.16012)].
scientific article; zbMATH DE number 1139679