Character Formulae in Category O for Periplectic Lie Superalgebras(1/2)

Project Details

Description

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.
StatusFinished
Effective start/end date1/10/1931/07/20

Keywords

  • Periplectic Lie superalgebra
  • category O
  • character formula
  • projective module
  • tilting module.

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