1999/08/14 by Martin Welk, Welk, Martin
Mathematics · #17B37 #46L87 #81R50 #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA #msc:17B37 #msc:46L87 #msc:81R50
paper · pdf · doi:10.48550/arxiv.math/9908069
20 pages; Latex2e, amsmath, amssymb
arxiv created 1999/08/14 · arxiv updated 2009/11/30
We investigate covariant first order differential calculi on the quantum complex projective spaces CPqN-1 which are quantum homogeneous spaces for the quantum group SUq(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 Sq2N-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 CPqN-1: 1. restriction of covariant first order differential calculi from Sq2N-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 CPqN-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 CPqN-1 are also obtained by restriction from calculi on Sq2N-1 as well as from calculi on the quantum group SUq(N).