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SOME HOMOLOGICAL PROPERTIES OF CATEGORY $\boldsymbol {\mathcal {O}}$ FOR LIE SUPERALGEBRAS

Published online by Cambridge University Press:  21 January 2022

CHIH-WHI CHEN*
Affiliation:
Department of Mathematics, National Central University, Zhongli District, Taoyuan City, Taiwan
VOLODYMYR MAZORCHUK
Affiliation:
Department of Mathematics, Uppsala University, Box 480, SE-75106 Uppsala, Sweden e-mail: mazor@math.uu.se

Abstract

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\Delta (\lambda )$ to be such that every nonzero homomorphism from another Verma supermodule to $\Delta (\lambda )$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak {pe} (n)$ and, furthermore, to reduce the problem of description of $\mathrm {Ext}^1_{\mathcal O}(L(\mu ),\Delta (\lambda ))$ for $\mathfrak {pe} (n)$ to the similar problem for the Lie algebra $\mathfrak {gl}(n)$ . Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\mathcal O^{\mathfrak {p}}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak {pe} (n)$ and the orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$ .

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Anthony Henderson

The first author is partially supported by MoST grant 108-2115-M-008-018-MY2. For the second author, the research was partially supported by the Swedish Research Council and Göran Gustafssons Stiftelse.

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