Tilting modules for the current algebra of a simple Lie algebra

TL;DR

This paper studies tilting modules in the category of level zero representations of current algebras associated with simple Lie algebras, establishing their structure and properties.

Contribution

It introduces the construction of indecomposable tilting modules and characterizes their decomposition in the current algebra setting.

Findings

Canonical filtrations are characterized for type A Lie algebras.

Indecomposable tilting modules are constructed explicitly.

Any tilting module decomposes into a direct sum of indecomposables.

Abstract

The category of level zero representations of current and affine Lie algebras shares many of the properties of other well-known categories which appear in Lie theory and in algebraic groups in characteristic p and in this paper we explore further similarities. The role of the standard and co-standard module is played by the finite-dimensional local Weyl module and the dual of the infinite-dimensional global Weyl module respectively. We define the canonical filtration of a graded module for the current algebra. In the case when $\mathfrak g$ is of type $\mathfrak{sl}_{n+1}$ we show that the well-known necessary and sufficient homological condition for a canonical filtration to be a good (or a $\nabla$-filtration) also holds in our situation. Finally, we construct the indecomposable tilting modules in our category and show that any tilting module is isomorphic to a direct sum of indecomposable tilting modules.