維度(4,4)的正交糾紐李超代數的對稱張量

Abstract

正交糾紐李超代數 osp 可用微分算子加以實現， 齊次多項式空間在它的作用下是封閉的，也就是說， 齊次多項式空間是 osp-模。 本論文旨在探討齊次多項式空間是否能分解成不可約 osp-模的直合。 我們得到的結論是對於任何奇數次的齊次多項式空間而言，都是對的。 至於偶數，在二次時即已不成立， 因而對於任何偶數次的齊次多項式空間亦不成立， 原因是它們必包含有一子模同構於二次齊次多項式空間。然而， 任意偶數次齊次多項式空間的分解問題仍未得到解決。

Ortho-symplectic Lie superalgebra osp can be realized as differential operators and homogeneous polynomial space is closed under its action, that is, homogeneous polynomial space is an osp-module. Our thesis is to study whether or not homogeneous polynomial space can be reduced to a direct sum of irreducible osp-modules. Our conclusion is for any odd homogeneous polynomial space, the answer is yes. For even, the answer is no in the case of degree 2, and therefore invalid for any even homogeneous polynomial space since it must contain a submodule isomorphic to degree 2 homogeneous polynomial space. However, a complete decomposition of arbitrary even homogeneous polynomial space has not been reached yet.