Differential Calculus on Quantum Complex Grassmann Manifolds I: Construction

Covariant first order differential calculus over quantum complex Grassmann manifolds is considered. It is shown by a Pusz–Woronowicz type argument that under restriction to calculi close to classical Kähler differentials there exist exactly two such calculi for the homogeneous coordinate ring. Complexification and localization procedures are used to induce covariant first order differential calculi over quantum Grassmann manifolds. It is shown that these differential calculi behave in many respects as their classical counterparts. As an example the q-deformed Chern character of the tautological bundle is constructed. Covariant first order differential calculus is a concept first introduced by S. L. Woronowicz [Wor87], [Wor89] to generalize the notion of differential form from commutative algebra to quantum groups and quantum spaces. The task to find well behaved analogues of differential forms for the noncommutative deformed coordinate algebras appearing in the framework of quantum groups is still of considerable interest [Her98], [Sch99], [SV98]. In [PW89] W. Pusz and S.L. Woronowicz proved that for the quantum vector space of dimension ≥ 3 there exist exactly two differential calculi freely generated by the differentials of the generators. In [Pod92] P. Podleś classified differential structures on the quantum 2-sphere S qc which have certain properties similar to classical differential forms. It turned out, that only in the so called quantum subgroup case c = 0 such a differential calculus exists and is then uniquely determined. Podleś quantum 2-sphere is an example of a q-deformed Grassmann manifold. The undeformed Grassmann manifold Gr(r,N) of r-dimensional subspaces in C is a projective algebraic variety. It is well known [CP94] that its homogeneous coordinate ring O(Vr) can be q-deformed to a quantum space Oq(Vr). On the other hand M. Noumi, M. S. Dijkhuizen and T. Sugitani introduced a large class of quantum Grassmannians [NDS97]. Here only the quantum subgroup case is considered and will be denoted by Oq(Gr(r,N)). The aim of this paper is to construct a canonical covariant first order differential calculus over Oq(Gr(r,N)) and investigate its properties. The q-deformed coordinate rings Oq(Vr) and Oq(Gr(r,N)) are closely related. More explicitly we consider the complexification of Oq(Vr), an algebra obtained by adding complex conjugate elements. This complexification has a distinguished