The Locally Finite Part of the Dual Coalgebra of Quantized Irreducible Flag Manifolds

The notion of locally finite part of the dual coalgebra of certain quantized coordinate rings is introduced. In the case of irreducible flag manifolds this locally finite part is shown to coincide with a natural quotient coalgebra U of Uq(g). On the way the coradical filtration of U is determined. A graded version of the duality between U and the quantized coordinate ring is established. This leads to a natural construction of several examples of quantized vector spaces. As an application covariant first order differential calculi on quantized irreducible flag manifolds are classified.