sl（2）圈代数有限维最高权表示的不可约准则及可约表示的维数
Irreducibility criterion for a finite-dimensional highest weight representation of the sl(2) loop algebra and the dimensions of reducible representations
摘要：
给出了$sl_2$循环代数的有限维最高权表示不可约的充要条件。特别地，对于具有最高权的退化参数的最高权表示，我们可以明确地确定它是否是不可约的。我们还提出了一个算法，用于构建有限维最高权表示与给定的最高权。我们提出了一个猜想，即所有具有相同最高权的最高权表示都可以由该算法构造出来。对于一些例子，我们明确地证明了这个猜想。这一结果对于分析与量子群单位根表示相关的可积格点模型的谱，特别是分析与$sl_2$圈代数相关的XXZ自旋链在单位根处的谱简并性，将是有用的.
英文摘要：
We present a necessary and sufficient condition for a finite-dimensional highest weight representation of the $sl_2$ loop algebra to be irreducible. In particular, for a highest weight representation with degenerate parameters of the highest weight, we can explicitly determine whether it is irreducible or not. We also present an algorithm for constructing finite-dimensional highest weight representations with a given highest weight. We give a conjecture that all the highest weight representations with the same highest weight can be constructed by the algorithm. For some examples we show the conjecture explicitly. The result should be useful in analyzing the spectra of integrable lattice models related to roots of unity representations of quantum groups, in particular, the spectral degeneracy of the XXZ spin chain at roots of unity associated with the $sl_2$ loop algebra.
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