在李代數的研究上，表現理論扮演著很重要的角色，而如何去分類一個李代數的表現
是一件困難且重要的任務。這個問題目前只有在 sl_2 上的不可分表現由 Block 解出。
Harish-Chandra 在 semisimple 李代數上，對其所有的 central character 提出了重要結
果。而 Duflo 延續了 Harish-Chandra 的理論，證明了 semisimple 李代數的 enveloping
algebra 上的 primitive ideals ，會是某些 simple module 的 annihilator。
但在 semisimple 李代數的 Takiff 代數上，這些理論卻不一定正確。Chen 和 Wang 在其
論文中提出一個不滿足 Harish-Chandra 的反例。而本篇論文主要在探討：在哪些李代數
的 Takiff 代數上，無法滿足 Harish-Chandra 和 Duflo 的理論，其中我們會使用 Molev
和 Tauvel 的理論來證明。

To study Lie algebras, the theory of representation plays a crucial role. The problem
of classification of representation for a certain Lie algebra is a challenging and important task, for which the solution exists only for irreducible representations for sl_2 due to
Block. The results established by Harish-Chandra provide that every central character, a
1-dimensional representation for the center of enveloping algebra, is the central character
of certain simple modules, for all semisimple Lie algebras. Extending the work of Harish-
Chandra, Duflo proved that every primitive ideal of enveloping algebra is the annihilator
of a certain simple module. In Takiff algebras of a semisimple algebra, however, the theo-
rem may not hold, since Chen and Wang pointed out that in the 1-st Takiff algebra of sl2,
there exists a central character, which does not satisfy the statement of Harish-Chandra’s
theorem. In this thesis, our work is to demonstrate that this condition doesn’t hold for
the Takiff algebra of certain types of Lie algebras, using the theorem of Molev and Tauvel.

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