Categories $\mathcal{O}$ for Root-Reductive Lie Algebras: II. Translation Functors and Tilting Modules

TL;DR

This paper extends the theory of categories al for root-reductive Lie algebras by developing translation functors and tilting modules, establishing equivalences between blocks and introducing universal tilting objects.

Contribution

It introduces translation functors and tilting modules in categories al for root-reductive Lie algebras, and constructs universal tilting objects analogous to finite-dimensional cases.

Findings

Equivalence of subcategories via translation functors for certain blocks.

Definition of tilting objects within the categories al.

Construction of universal tilting objects D(al) similar to finite-dimensional theory.

Abstract

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.