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

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  21 January 2022

CHIH-WHI CHEN Open the ORCID record for CHIH-WHI CHEN [Opens in a new window]  and
VOLODYMYR MAZORCHUK
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: [email protected]
*
e-mail: [email protected]
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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 $\mathfrak {osp}(2|2n)$ .

Keywords

Lie algebraLie superalgebramoduleprojective dimensionsocle

MSC classification

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 17B55: Homological methods in Lie (super)algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 114 , Issue 1 , February 2023 , pp. 50 - 77
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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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