This is the second paper of a series of papers on a version of categories $\mathcal{ O}$ for root-reductive Lie algebras. Let $\mathfrak{ g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{ K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{ b}$ containing a splitting maximal toral subalgebra $\mathfrak{ h}$. For some pairs of blocks $\overline{ \mathcal{ O}}[\lambda]$ and $\overline{ \mathcal{ O}}[\mu]$, the subcategories whose objects have finite length are equivalence via functors obtained by the direct limits of translation functors. Tilting objects can also be defined in $\overline{ \mathcal{ O}}$. There are also universal tilting objects $D(\lambda)$ in parallel to the finite-dimensional cases.
Categories: math.RT