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Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category
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2018-01-01 12:13  »ª¶«Ê¦·¶´óÑ§

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We show that for any singular dominant integral weight $\lambda$ of a complex semisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra
$B_\lambda ^{\mathfrak{p}}$ of any projective-injective module in the parabolic BGG category $\mathcal{O}_\lambda^{\mathfrak{p}}$ 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 of any projective-injective module in $\mathcal{O}_\lambda^{\mathfrak{p}}$ equips with a homogeneous non-degenerate symmetrizing form. This is a joint work with Ngau Lam.