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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Category $\mathcal O$: Quivers and endomorphism rings of projectives
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by Catharina Stroppel PDF
Represent. Theory 7 (2003), 322-345 Request permission


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: and
  • 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
  • © Copyright 2003 American Mathematical Society
  • Journal: Represent. Theory 7 (2003), 322-345
  • MSC (2000): Primary 17B10, 16G20
  • DOI:
  • MathSciNet review: 2017061