Dual polar graphs, the quantum algebra U_q(sl_2), and Leonard systems of dual q-Krawtchouk type
Dual polar graphs, the quantum algebra U_q(sl_2), and Leonard systems of dual q-Krawtchouk type
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DOI：
10.1016/j.laa.2012.08.016
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发表时间：
2012-05
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通讯作者：
Chalermpong Worawannotai
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作者：
Chalermpong Worawannotai
本文考虑以下三个对象是如何联系的：（i）对偶极图;（ii）量子代数Uq（sl ~ 2）;（iii）对偶q-Krawtchouk型伦纳德系统。为了方便起见，我们首先描述（ii）和（iii）是如何相关的。对于给定的对偶q-Krawtchouk型伦纳德系统，我们得到了其基础向量空间上的两个Uq（sl ~ 2）-模结构.我们现在描述（i）和（iii）是如何相关的。设Γ表示对偶极图。固定Γ的一个顶点x，设T=T（x）表示相应的子成分代数。根据定义，T由Γ的邻接矩阵A和对应于x的称为对偶邻接矩阵的某个对角矩阵A*=A*（x）生成。通过构造，代数T是半单的。我们证明了对于每个不可约T-模W，A和A* 对W的限制诱导出一个对偶q-Krawtchouk型的伦纳德系统.我们现在描述（i）和（ii）是如何相关的。得到了Γ的标准模上的两个Uq（sl ~ 2）-模结构.我们描述了这两个Uq（sl 2）-模结构是如何相关的。每一个Uq（sl ~ 2）-模结构都导出一个C-代数同态Uq（sl ~ 2）→T。我们表明，在每一种情况下，T是由图像连同中心的T。利用Γ的组合学，我们得到了T沿着的生成元集L，F，R，K，这些生成元满足吸引关系.
In this paper we consider how the following three objects are related: (i) the dual polar graphs; (ii) the quantum algebra Uq(sl2); (iii) the Leonard systems of dual q-Krawtchouk type. For convenience we first describe how (ii) and (iii) are related. For a given Leonard system of dual q-Krawtchouk type, we obtain two Uq(sl2)-module structures on its underlying vector space. We now describe how (i) and (iii) are related. Let Γ denote a dual polar graph. Fix a vertex x of Γ and let T=T(x) denote the corresponding subconstituent algebra. By definition T is generated by the adjacency matrix A of Γ and a certain diagonal matrix A*=A*(x) called the dual adjacency matrix that corresponds to x. By construction the algebra T is semisimple. We show that for each irreducible T-module W the restrictions of A and A*to W induce a Leonard system of dual q-Krawtchouk type. We now describe how (i) and (ii) are related. We obtain two Uq(sl2)-module structures on the standard module of Γ. We describe how these two Uq(sl2)-module structures are related. Each of these Uq(sl2)-module structures induces a C-algebra homomorphism Uq(sl2)→T. We show that in each case T is generated by the image together with the center of T. Using the combinatorics of Γ we obtain a generating set L,F,R,K of T along with some attractive relations satisfied by these generators.