The locally finite part of the dual coalgebra of quantized irreducible flag manifolds

Abstract

For quantized irreducible flag manifolds the locally finite part of the dual coalgebra is shown to coincide with a natural quotient coalgebra $\overline{U}$ of $U_q ( \mathfrak{g} )$. On the way the coradical filtration of $\overline{U}$ is determined. A graded version of the duality between $\overline{U}$ 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.