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Category $\mathcal O$: Quivers and endomorphism rings of projectives


Author: Catharina Stroppel
Journal: Represent. Theory 7 (2003), 322-345
MSC (2000): Primary 17B10, 16G20
DOI: https://doi.org/10.1090/S1088-4165-03-00152-3
Published electronically: August 8, 2003
MathSciNet review: 2017061
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Abstract: We describe an algorithm for computing quivers of category $\mathcal O$ of a finite dimensional semisimple Lie algebra. The main tool for this is Soergel's description of the endomorphism ring of the antidominant indecomposable projective module of a regular block as an algebra of coinvariants. We give explicit calculations for root systems of rank 1 and 2 for regular and singular blocks and also quivers for regular blocks for type $A_3$.

The main result in this paper is a necessary and sufficient condition for an endomorphism ring of an indecomposable projective object of $\mathcal O$ to be commutative. We give also an explicit formula for the socle of a projective object with a short proof using Soergel's functor $\mathbb V$ and finish with a generalization of this functor to Harish-Chandra bimodules and parabolic versions of category $\mathcal O$.


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Additional Information

Catharina Stroppel
Affiliation: Mathematische Fakultät, Universität Freiburg, Germany
Email: stroppel@imf.au.dk and cs93@le.ac.uk

DOI: https://doi.org/10.1090/S1088-4165-03-00152-3
Keywords: Category $\mathcal O$, projectives, quivers, semisimple Lie algebras, Kazhdan-Lusztig
Received by editor(s): January 7, 2002
Received by editor(s) in revised form: April 7, 2003, and June 10, 2003
Published electronically: August 8, 2003
Additional Notes: The author was partially supported by EEC TMR-Network ERB FMRX-CT97-0100
Article copyright: © Copyright 2003 American Mathematical Society