论文标题
最小有限的最高权重理论$ w $ -superalgebras和相关的惠特克类别
Highest weight theory for minimal finite $W$-superalgebras and related Whittaker categories
论文作者
论文摘要
令$ \ mathfrak {g} = \ mathfrak {g} _ {\ bar0}+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {g} _ {\ bar1} $是$ \ mathbb {c} $的基本经典superalgebra，yathbb {c} $，以及$ e = e = e_trak in \ math. $ \ mathfrak {g} $的最小根。设置$ u（\ mathfrak {g}，e）$作为与Pair $（\ Mathfrak {g}，e）$关联的最小有限$ W $ superalgebras。在本文中，我们研究了$ u（\ mathfrak {g}，e）$的最高权重理论，介绍了Verma模块，并通过由重量和水平对组成的参数集对有限维不可减少模块进行了完整的同构分类。可以通过$ \ mathfrak {osp}（1 | 2）$或$ \ m athfrak {sl}（2）$从惠特克模块的抛物线诱导中进一步描述那些Verma模块，这取决于$ \ \ textsf {r}的均等范围，这然后，我们介绍并研究了$ u（\ mathfrak {g}，e）$的BGG类别$ \ Mathcal {O} $，建立了最高权重理论，作为Brundan-Goodwin-Kleshchev和Lossv的有限$ W $ -Algebras作品的作品。 与非usper案相比，这里的显着差异在于$ \ textsf {r} $奇怪的情况，这是一种全新的现象。那里的困难和复杂的计算来自那里。
Let $\mathfrak{g}=\mathfrak{g}_{\bar0}+\mathfrak{g}_{\bar1}$ be a basic classical Lie superalgebra over $\mathbb{C}$, and $e=e_θ\in\mathfrak{g}_{\bar0}$ with $-θ$ being a minimal root of $\mathfrak{g}$. Set $U(\mathfrak{g},e)$ to be the minimal finite $W$-superalgebras associated with the pair $(\mathfrak{g},e)$. In this paper we study the highest weight theory for $U(\mathfrak{g},e)$, introduce the Verma modules and give a complete isomorphism classification of finite-dimensional irreducible modules, via the parameter set consisting of pairs of weights and levels. Those Verma modules can be further described via parabolic induction from Whittaker modules for $\mathfrak{osp}(1|2)$ or $\mathfrak{sl}(2)$ respectively, depending on the detecting parity of $\textsf{r}:=\dim\mathfrak{g}(-1)_{\bar1}$. We then introduce and investigate the BGG category $\mathcal{O}$ for $U(\mathfrak{g},e)$, establishing highest weight theory, as a counterpart of the works for finite $W$-algebras by Brundan-Goodwin-Kleshchev and Losev, respectively. In comparison with the non-super case, the significant difference here lies in the situation when $\textsf{r}$ is odd, which is a completely new phenomenon. The difficulty and complicated computation arise from there.
