Covariant first order differential calculus on quantum projective spaces

Abstract

We investigate covariant first order differential calculi on the quantum complex projective spaces CP_q^{N-1} which are quantum homogeneous spaces for the quantum group SU_q(N). Hereby, one more well-studied example of covariant first order differential calculus on a quantum homogeneous space is given. Since the complex projective spaces are subalgebras of the quantum spheres S_q^{2N-1} introduced by Vaksman and Soibelman, we get also an example of the relations between covariant differential calculus on two closely related quantum spaces. Two approaches are combined in obtaining covariant first order differential calculi on CP_q^{N-1}: 1. restriction of covariant first order differential calculi from S_q^{2N-1}; 2. classification of calculi under appropriate constraints, using methods from representation theory. The main result is that under three reasonable settings of dimension constraints, covariant first order differential calculi on CP_q^{N-1} exist and are (for N >= 6) uniquely determined. This is a clear difference as compared to the case of the quantum spheres where several parametrical series of calculi exist. For two of the constraint settings, the covariant first order calculi on CP_q^{N-1} are also obtained by restriction from calculi on S_q^{2N-1} as well as from calculi on the quantum group SU_q(N).