Irreducible representations of untwisted affine Kac-Moody algebras
Irreducible representations of untwisted affine Kac-Moody algebras
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发表时间：
2013-05
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通讯作者：
Xiangqian Guo;K. Zhao
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作者：
Xiangqian Guo;K. Zhao
本文在非扭曲仿射Kac-Moody代数$\ widdetilde {\mathfrak{g}}$上构造了一类新的不可约模，它推广并包含了最高权模和Whittaker模。这些模块允许我们得到不可约$\ widdetilde {\mathfrak{g}}$的完全分类，其中$\ widdetilde {\mathfrak{n}}_+$中的每个根向量的作用是局部有限的，其中$\ widdetilde {\mathfrak{n}}_+$是$\ widdetilde {\mathfrak{g}}$的局部幂零子代数（或正部分）。并确定了两个不可约$\ widdetilde {\mathfrak{g}}$-模同构的充分必要条件。在论文的第二部分，我们利用“移位技术”得到了不可约可积环$\ widdetilde {\mathfrak{g}}$-模与不可约可积最高权$\ widdetilde {\mathfrak{g}}$-模的张量积简单的充要条件。这个张量积问题最初是由Chari和Pressley在28年前研究的。
In this paper we construct a class of new irreducible modules over untwisted affine Kac-Moody algebras $\widetilde{\mathfrak{g}}$, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to obtain a complete classification of irreducible $\widetilde{\mathfrak{g}}$-modules on which the action of each root vector in $\widetilde{\mathfrak{n}}_+$ is locally finite, where $\widetilde{\mathfrak{n}}_+$ is the locally nilpotent subalgebra (or positive part) of $\widetilde{\mathfrak{g}}$. The necessary and sufficient conditions for two such irreducible $\widetilde{\mathfrak{g}}$-modules to be isomorphic are also determined. In the second part of the paper, we use the "shifting technique" to obtain a necessary and sufficient condition for the tensor product of irreducible integrable loop $\widetilde{\mathfrak{g}}$-modules and irreducible integrable highest weight $\widetilde{\mathfrak{g}}$-modules to be simple. This tensor product problem was originally studied by Chari and Pressley 28 years ago.