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Highest weight $ \mathfrak{sl}_2$-categorifications II: Structure theory


Author: Ivan Losev
Journal: Trans. Amer. Math. Soc. 367 (2015), 8383-8419
MSC (2010): Primary 18D99, 05E10; Secondary 16G99, 17B10, 20G05
Published electronically: April 9, 2015
MathSciNet review: 3403059
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Abstract: This paper continues the study of highest weight categorical $ \mathfrak{sl}_2$-actions initiated in part I. We start by refining the definition given there and showing that all examples considered in part I are also highest weight categorifications in the refined sense. Then we prove that any highest weight $ \mathfrak{sl}_2$-categorification can be filtered in such a way that the successive quotients are so-called basic highest weight $ \mathfrak{sl}_2$-categorifications. For a basic highest weight categorification we determine minimal projective resolutions of standard objects. We use this, in particular, to examine the structure of tilting objects in basic categorifications and to show that the Ringel duality is given by the Rickard complex. We apply some of these structural results to categories $ \mathcal {O}$ for cyclotomic Rational Cherednik algebras.


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Additional Information

Ivan Losev
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: i.loseu@neu.edu

DOI: https://doi.org/10.1090/tran/6311
Received by editor(s): September 15, 2012
Received by editor(s) in revised form: May 29, 2013, and September 12, 2013
Published electronically: April 9, 2015
Additional Notes: This work was supported by the NSF grants DMS-0900907 and DMS-1161584
Article copyright: © Copyright 2015 American Mathematical Society