Category 𝒪: Quivers and endomorphism rings of projectives

Stroppel, Catharina

doi:10.1090/S1088-4165-03-00152-3

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

Represent. Theory 7 (2003), 322-345

DOI: https://doi.org/10.1090/S1088-4165-03-00152-3

Published electronically: August 8, 2003

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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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