论文标题
类别$ \ MATHCAL O $的一些同源特性用于Lie Superalgebras
Some homological properties of category $\mathcal O$ for Lie superalgebras
论文作者
论文摘要
对于I型的经典谎言超级乳房，我们为Verma Supermodule $δ（λ）$提供必要的条件，以使每个非零同型从另一个Verma Supermodule到$δ（λ）$都是iNjementive。这适用于描述cokernel的socle，其中包含verma超模型超过周期性的超gebras $ \ mathfrak {pe} {pe}（n）$，此外，减少了$ \ mathrm {extrm {extrm {extrm {extrm {ext}^1 _ _ {\ mathcal o}的描述问题$ \ mathfrak {pe}（n）$ to lie代数$ \ mathfrak {gl}（n）$的类似问题。 此外，我们研究了抛物线超级模型的射质超模型的投影和注入尺寸$ \ Mathcal o^{\ Mathfrak P} $用于经典的Lie Superalgebras。特别是，我们完全确定了周期性超模型的这些维度超级模块superalgebra $ \ mathfrak {pe}（n）$和Ortho-Splectic lie superalgebra $ \ mathfrak {osp {osp}（osp}（2 | 2n）$。
For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $Δ(λ)$ to be such that every non-zero homomorphism from another Verma supermodule to $Δ(λ)$ 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 $\mathrm{Ext}^1_{\mathcal O}(L(μ),Δ(λ))$ 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 ortho-symplectic Lie superalgebra $\mathfrak{osp}(2|2n)$.
