Abstract
We describe an algorithm for computing quivers of category O \mathcal O of a finite dimensional semisimple Lie algebra. The main tool for this is Soergelâs 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 A_3 . The main result in this paper is a necessary and sufficient condition for an endomorphism ring of an indecomposable projective object of O \mathcal O to be commutative. We give also an explicit formula for the socle of a projective object with a short proof using Soergelâs functor V \mathbb V and finish with a generalization of this functor to Harish-Chandra bimodules and parabolic versions of category O \mathcal O .
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More From: Representation Theory of the American Mathematical Society
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