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

SIGMA 11 (2015), 016, 17 pages      arXiv:1409.5717      https://doi.org/10.3842/SIGMA.2015.016
Contribution to the Special Issue on New Directions in Lie Theory

Extension Fullness of the Categories of Gelfand-Zeitlin and Whittaker Modules

Kevin Coulembier a and Volodymyr Mazorchuk b
a) Department of Mathematical Analysis, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
b) Department of Mathematics, Uppsala University, Box 480, SE-751 06, Uppsala, Sweden

Received September 25, 2014, in final form February 20, 2015; Published online February 24, 2015

Abstract
We prove that the categories of Gelfand-Zeitlin modules of $\mathfrak{g}=\mathfrak{gl}_n$ and Whittaker modules associated with a semi-simple complex finite-dimensional algebra $\mathfrak{g}$ are extension full in the category of all $\mathfrak{g}$-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand-Zeitlin and Whittaker modules.

Key words: extension fullness; Gelfand-Zeitlin modules; Whittaker modules; Yoneda extensions; homological dimension.

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