Central Characters of Takiff Algebra in Types A, B, C, D

Please use this identifier to cite or link to this item: https://ir.lib.ncu.edu.tw/handle/987654321/98512

Title:	Central Characters of Takiff Algebra in Types A, B, C, D
Authors:	劉泠瑋;Liu, Ling-Wei
Contributors:	數學系
Keywords:	李代數;Takiff 代數;Lie algebra;Central character;Takiff algebra
Date:	2025-07-07
Issue Date:	2025-10-17 12:52:15 (UTC+8)
Publisher:	國立中央大學
Abstract:	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.
Appears in Collections:	[Graduate Institute of Mathematics] Electronic Thesis & Dissertation

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