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- Title
SOME HOMOLOGICAL PROPERTIES OF CATEGORY $\boldsymbol {\mathcal {O}}$ FOR LIE SUPERALGEBRAS.
- Authors
CHEN, CHIH-WHI; MAZORCHUK, VOLODYMYR
- 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 $\mathfrak {osp}(2|2n)$.
- Subjects
MODULES (Algebra); LIE algebras; LIE superalgebras; HOMOMORPHISMS; ABELIAN categories
- Publication
Journal of the Australian Mathematical Society, 2023, Vol 114, Issue 1, p50
- ISSN
1446-7887
- Publication type
Article
- DOI
10.1017/S1446788721000239