奇特型李超代數的表現理論

Abstract

本博士論文主要研究奇特型李超代數的表現理論， 並分成三個部分。 首先，我們研究 Periplectic 李超代數的（整權）有限維模範疇， 並且得到了 BGG 型之互反律。 特別地， 僅存在有限個不可分解的區塊。 我們亦計算了秩為2 與3 之Periplectic 李超代數的不可約特徵標， 並得到了對於秩為2,3,4 之Periplectic 李超代數的區塊描述。 在第二個部分， 我們對 Queer 李超代數的BGG 範疇發展了約化方法。 我們也建立了在某個 Queer 李超代數的極大拋物子範疇與一般線性李超代數的有限維模範疇區塊之間的等價。 在最後的章節，我們對Queer 李超代數的極大拋物子範疇證明了 Kazhdan-Lusztig 型猜想。 作為應用， 我們得到與一般線性李超代數之模範疇情況相似的封閉不可約特徵標公式。

In this dissertation, we study the representation theory of strange Lie superalgebras. It is divided into three parts. In the first part, we study categories of finite-dimensional modules over the periplectic Lie superalgebras $mathfrak{p}(n)$ and obtain a BGG type reciprocity. In particular, these categories have only finitely-many blocks. We also compute the characters for irreducible modules over periplectic Lie superalgebras of ranks $2$ and $3$, and obtain explicit description of the blocks for ranks $2$, $3$, and $4$. In the second part, we develop a reduction procedure which provides an equivalence from an arbitrary block of the BGG category for the queer Lie superalgebra $mathfrak{q}(n)$ to a block with weights in $Lambda_{{ell_1},s_{1}}(n_1) imes cdots imes Lambda_{{ell_k},s_{k}}(n_{k})$ (see, Theorem ef{FirstMainThm}) for a BGG category of finite direct sum of queer Lie superalgebras. The descriptions of blocks are given as well. We also establish equivalences between certain maximal parabolic subcategories for $mathfrak{q}(n)$ and blocks of atypicality-one of the category of finite-dimensional modules for $mathfrak{gl}(ell|n-ell)$, where $ell leq n$. In the third part, we establish a maximal parabolic version of the Kazhdan-Lusztig conjecture cite[Conjecture 5.10]{CKW} for the BGG category $mathcal{O}_{k,zeta}$ of $mathfrak{q}(n)$-modules of ``$pm zeta$-weights', where $kleq n$ and $zetainCsetminushf $. As a consequence, the irreducible characters of these $mathfrak q(n)$-modules in this maximal parabolic category are given by the Kazhdan-Lusztig polynomials of type $A$ Lie algebras. As an application, closed character formulas for a class of $mathfrak q(n)$-modules resembling polynomial and Kostant modules of the general linear Lie superalgebras are obtained.