### Abstract

We show that for any singular dominant integral weight λ of a complex semisimple Lie algebra g, the endomorphism algebra B of any projective-injective module of the parabolic BGG category O
_{λ}
^{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 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}
^{p}
of the basic projective-injective module of O
_{λ}
^{p}
is a symmetric algebra.

Original language | English |
---|---|

Pages (from-to) | 173-201 |

Number of pages | 29 |

Journal | Journal of Algebra |

Volume | 515 |

DOIs | |

Publication status | Published - 2018 Dec 1 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

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**Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category.** / Hu, Jun; Lam, Ngau.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category

AU - Hu, Jun

AU - Lam, Ngau

PY - 2018/12/1

Y1 - 2018/12/1

N2 - We show that for any singular dominant integral weight λ of a complex semisimple Lie algebra g, the endomorphism algebra B of any projective-injective module of the parabolic BGG category O λ 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 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 p of the basic projective-injective module of O λ p is a symmetric algebra.

AB - We show that for any singular dominant integral weight λ of a complex semisimple Lie algebra g, the endomorphism algebra B of any projective-injective module of the parabolic BGG category O λ 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 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 p of the basic projective-injective module of O λ p is a symmetric algebra.

UR - http://www.scopus.com/inward/record.url?scp=85053125752&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053125752&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2018.08.020

DO - 10.1016/j.jalgebra.2018.08.020

M3 - Article

VL - 515

SP - 173

EP - 201

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -