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Homological invariants in category $ \mathcal{O}$ for the general linear superalgebra


Authors: Kevin Coulembier and Vera Serganova
Journal: Trans. Amer. Math. Soc. 369 (2017), 7961-7997
MSC (2010): Primary 17B10, 16E30, 17B55
DOI: https://doi.org/10.1090/tran/6891
Published electronically: May 5, 2017
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Abstract: We study three related homological properties of modules in the BGG category  $ \mathcal {O}$ for basic classical Lie superalgebras, with specific focus on the general linear superalgebra. These are the projective dimension, associated variety and complexity. We demonstrate connections between projective dimension and singularity of modules and blocks. Similarly we investigate the connection between complexity and atypicality. This creates concrete tools to describe singularity and atypicality as homological, and hence categorical, properties of a block. However, we also demonstrate how two integral blocks in category  $ \mathcal {O}$ with identical global categorical characteristics of singularity and atypicality will generally still be inequivalent. This principle implies that category  $ \mathcal {O}$ for $ \mathfrak{gl}(m\vert n)$ can contain infinitely many non-equivalent blocks, which we work out explicitly for $ \mathfrak{gl}(3\vert 1)$. All of this is in sharp contrast with category  $ \mathcal {O}$ for Lie algebras, but also with the category of finite dimensional modules for superalgebras. Furthermore we characterise modules with finite projective dimension to be those with trivial associated variety. We also study the associated variety of Verma modules. To do this, we also classify the orbits in the cone of self-commuting odd elements under the action of an even Borel subgroup.


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Additional Information

Kevin Coulembier
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia

Vera Serganova
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720

DOI: https://doi.org/10.1090/tran/6891
Keywords: Category $\mathcal{O}$, general linear superalgebra, projective dimension, equivalence of blocks, associated variety, complexity, Kazhdan-Lusztig theory
Received by editor(s): April 8, 2015
Received by editor(s) in revised form: November 4, 2015, and December 12, 2015
Published electronically: May 5, 2017
Additional Notes: The first author is a Postdoctoral Fellow of the Research Foundation-Flanders (FWO)
The second author was partially supported by NSF grant 1303301
Article copyright: © Copyright 2017 American Mathematical Society