We introduce and study a category of representations of the Borel algebra associated with a quantum loop algebra of non-twisted type. We construct fundamental representations for this category as a limit of the Kirillov–Reshetikhin modules over the quantum loop algebra and establish explicit formulas for their characters. We prove that general simple modules in this category are classified by n-tuples of rational functions in one variable which are regular and non-zero at the origin but may have a zero or a pole at infinity.

We identify the Poisson boundary of the dual of the universal compact quantum group Au(F) with a measurable field of ITPFI (infinite tensor product of finite type I) factors.

We describe a general setting for the definition of semi-infinite cohomology of finite-dimensional graded algebras, and provide an interpretation of such cohomology in terms of derived categories. We apply this interpretation to compute semi-infinite cohomology of some modules over the small quantum group at a root of unity, generalizing an earlier result of Arkhipov (posed as a conjecture by B. Feigin).

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.