Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 030, 23 pages      arXiv:1307.3307
Contribution to the Special Issue on New Directions in Lie Theory

Tilting Modules in Truncated Categories

Matthew Bennett a and Angelo Bianchi b
a) Department of Mathematics, State University of Campinas, Brazil
b) Institute of Science and Technology, Federal University of São Paulo, Brazil

Received September 05, 2013, in final form March 17, 2014; Published online March 26, 2014; Rearrangement of Sections 2, 3 and 7, reference [5] updated, misprints corrected May 02, 2014

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 $\mathcal G(\Gamma')$ where $\Gamma' = P' \times J$, where $P'\subseteq P^+$ is saturated. Under certain natural conditions on $\Gamma'$, we note that $\mathcal G(\Gamma')$ admits full tilting modules.

Key words: current algebra; tilting module; Serre subcategory.

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