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SOME HOMOLOGICAL PROPERTIES OF CATEGORY $\boldsymbol {\mathcal {O}}$ FOR LIE SUPERALGEBRAS.
- Source :
-
Journal of the Australian Mathematical Society . Feb2023, Vol. 114 Issue 1, p50-77. 28p. - Publication Year :
- 2023
-
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 $\mathfrak {osp}(2|2n)$. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 14467887
- Volume :
- 114
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Journal of the Australian Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 161314101
- Full Text :
- https://doi.org/10.1017/S1446788721000239