本計劃的目的是研究periplectic李超代數的範疇 O。特別地，此李超代數的不可約特徵標問題目前還是未解的公開問題。唯一已知的是由本人與Kevin Coulembier (Sydney University)得到秩為二的情況。本計劃的首要目標是解決特徵標問題。主要的困難在於大部分的古典工具，包括中心分解的射影函子，在此情況都是失效的。因此需要全新的技巧來解決這個問題。我們主要採取的策略是利用新型態的轉移函子。在2016年左右，periplectic李超代數的假性Casimir算子被本人與彭勇寧（中央大學）構造出來。從此以後，periplectic李超代數的研究就有大幅度的進展。包括有限維模的特徵標、periplectic Brauer 及仿射periplectic Brauer、 以及他們的表現理論。我們首先研究，布於秩較小的periplectic李超代數之Verma 模的連接原理。然後利用Casimir 算子來發展轉移函子的一般性質。這將幫助我們解決秩較小情況中的所有特徵標問題。本計劃的第二個目的是給出範疇 O 中的簡化方法。 具體來說，在第一階段中只有考慮範疇O中整權的模。接著將利用這些結論來解決任意權的不可約特徵標問題。在A型李超代數的情況，這樣的簡化方法已經被所謂的拋物誘導方法完成。然而，這樣的方法並不適用於periplectic李超代數。因此我們需要建立任意秩的範疇O中的範疇化的唯一性定理。 ;The goal of this research project is to study the Bernstein-Gelfand-Gelfand category O for the periplectic Lie superalgebras. In particular, the problem of characters of irreducible modules in the category O is still open. The known case is rank 2 periplectic Lie superalgebra given in the previous work with Kevin Coulembier (Sydney University). Our first goal of this project is to solve character problem for the periplectic Lie superalgebras. The main difficulty is due to the fact that most classical tools including the projective functors of central blocks are unavailable. We expect that there will be new approaches to this problem. Namely, we expect to employ 'translation functors'.Around 2016, the so-called 'fake Casimir element' was constructed for the periplectic Lie superalgebras in my joint work with Yung-Ning Peng (National Central University). Since then the study of the periplectic Lie superalgebras has made great progress, including characters formulae of finite-dimensional modules, periplectic Brauer algebras, affine periplectic Brauer algebras and their representation theories. We shall firstly study the linkage principle in Verma modules for the periplectic Lie superalgebras in small rank. We will develop some general properties of translation functors using fake Casimir elements. This shall help us solve for all irreducible characters of periplectic Lie superalgebras in small rank.Our second goal is to give a reduction method of the category O. More precisely, in the first stage we only consider the category O of modules with integer weights. We shall solve for the character problem for arbitrary weights in terms of that of integral weight case. Such a reduction method of type A Lie superalgebras has been achieved by using the so-called parabolic induction method. However, this method does not work for the periplectic Lie superalgebras. Therefore we shall establish the uniqueness theorem of categorification of the categories O for periplectic Lie superalgebras in various ranks.
