Parabolic category $\mathcal O^{\mathfrak p}$ for periplectic Lie superalgebras $\mathfrak{pe}(n)$

Chen, Chih-Whi; Peng, Yung-Ning

doi:10.48550/arxiv.2002.10311

Cited by 1 publication

(3 citation statements)

References 26 publications

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“…Kac functor and homomorphisms between Verma supermodules. Using the same argument as in the proof of [CP,Lemma 4.3], we have the following useful lemma.…”

Section: Homomorphisms Between Verma Supermodulesmentioning

confidence: 99%

“…From [Co,Corollary 3.2.5] and [Co,Lemma 6.3 We first suppose that λ is typical. Then, by [CP,Theorem 4.6],…”

Section: (Dual) Verma Supermodules Over Osp(2|2n)mentioning

confidence: 99%

“…We define w λ 0 to be the longest element of W[λ] . For λ, ν ∈ Σ + p , it follows from[CP, Theorem B] (see also[CP, Theorem 3.2]) that O p λ = O p ν if and only if λ ∼ ν. As a consequence, there exists ν ∈ Irr O p λ which is dominant and regular.…”

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confidence: 99%

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Some Homological Properties of Category for Lie Superalgebras

CHEN

Mazorchuk

2022

J. Aust. Math. Soc.

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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 $\mathfrak {osp}(2|2n)$ .

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“…Kac functor and homomorphisms between Verma supermodules. Using the same argument as in the proof of [CP,Lemma 4.3], we have the following useful lemma.…”

Section: Homomorphisms Between Verma Supermodulesmentioning

confidence: 99%

“…From [Co,Corollary 3.2.5] and [Co,Lemma 6.3 We first suppose that λ is typical. Then, by [CP,Theorem 4.6],…”

Section: (Dual) Verma Supermodules Over Osp(2|2n)mentioning

confidence: 99%