Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category

2017/02/20 by Jun Hu, Hu, Jun, Ngau Lam +1 · 1 citation
Mathematics · #17B10 #20G05 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1702.05834

openalex publication_date 2017/02/20 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

We show that for any singular dominant integral weight λ of a complex semisimple Lie algebra \mathfrakg, the endomorphism algebra B of any projective-injective module of the parabolic BGG category O_λ^\mathfrakp is a symmetric algebra (as conjectured by Khovanov) extending the results of Mazorchuk and Stroppel for the regular dominant integral weight. Moreover, the endomorphism algebra B is equipped with a homogeneous (non-degenerate) symmetrizing form. In the appendix, there is a short proof due to K. Coulembier and V. Mazorchuk showing that the endomorphism algebra B_λ^\mathfrakp of the basic projective-injective module of O_λ^\mathfrakp is a symmetric algebra.

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