Tilting Modules in Truncated Categories

Abstract

We begin the study of a tilting theory in certain truncated categories of modules $\mathcal{G}(\Gamma )$ for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where $\Gamma =P^{+}\times J$, $J$ is an interval in $\mathbb{Z}$, and $P^{+}$ is the set of dominant integral weights of the simple Lie algebra. We use this to put a tilting theory on the category $\mathcal{G}({\Gamma }^{\prime })$ where ${\Gamma }^{\prime }=P^{\prime }\times J$, where $P^{\prime }\subseteq P^{+}$ is saturated. Under certain natural conditions on ${\Gamma }^{\prime }$, we note that $\mathcal{G}({\Gamma }^{\prime })$ admits full tilting modules.

v7: rearrangement of Sections 2, 3 and 7, reference [5] updated, misprints corrected

References in corpus (6)

• Current algebras, highest weight categories and quivers
• BGG reciprocity for current algebras
• Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas
• Minimal affinizations as projective objects
• Kirillov--Reshetikhin modules associated to G_2
• Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties