全几乎二部Leonard三元组的分类及构作

发布时间：2018-04-16 03:34

  本文选题：Leonard对 + Leonard三元组　； 参考：《河北师范大学》2017年博士论文

【摘要】：Leonard对和Leonard三元组等线性代数研究对象是研究结合方案的新理论.这一新理论统称为Terwilliger代数的表示理论,并且与李代数、量子群和数学物理有着紧密的联系.而以二部图和几乎二部图为组合背景的全二部Leonard对、全二部Leonard三元组以及全几乎二部Leonard对、全几乎二部Leonard三元组在Terwilliger代数的表示理论中显得更为重要且有意义.本文研究全几乎二部Leonard对和全几乎二部Leonard三元组的分类问题.对于给定的Bannai/Ito型全几乎二部Leonard对和给定的q-Racah型全几乎二部Leonard对,我们分别构作了相应的Leonard三元组,得到如下成果:1.证明了只有两种类型的全几乎二部Leonard对,分别是q-Racah型和Bannai/Ito型.利用量子包络代数Uq(so_3)的表示理论,在q不是单位根的条件下,给出了q-Racah型全几乎二部Leonard对的分类.2.利用量子包络代数U(so_3)的表示理论,在q不是单位根的条件下,给出了q-Racah型全几乎二部Leonard三元组的分类.3.设K是一个特征为零的代数闭域.选取K~(d+1)上一个特殊的Bannai/Ito型全几乎二部的Leonard对(A,A*),我们构作了所有的A~ε ∈ Mat_(d+1)(K),使得(A,A*,A~ε)构成K~(d+1)上的Leonard三元组.4.设K是一个特征为零的代数闭域.选取K~(d+1)上一个特殊的q-Racah型全几乎二部的Leonard对(A,A*),我们构作了所有的A~ε ∈ Mat_(d+1)(K),使得(A,A*,A~ε)构成K~(d+1)上的Leonard三元组.
[Abstract]:The research object of linear algebra such as Leonard pair and Leonard triple is a new theory to study the combined scheme.This new theory is called the representation theory of Terwilliger algebra and is closely related to lie algebra, quantum group and mathematical physics.The all-bipartite Leonard pairs, the all-bipartite Leonard triples and the all-almost bipartite Leonard pairs with bipartite and almost bipartite background are more important and meaningful in the representation theory of Terwilliger algebras.In this paper, we study the classification of all almost bipartite Leonard pairs and all almost bipartite Leonard triples.For a given Bannai/Ito type total almost bipartite Leonard pair and a given q-Racah type total almost bipartite Leonard pair, we construct corresponding Leonard triples respectively, and obtain the following result: 1.It is proved that there are only two types of total almost bipartite Leonard pairs, q-Racah type and Bannai/Ito type.By using the representation theory of quantum envelope algebra Uqso _ s _ 3, the classification of almost bipartite Leonard pairs of q-Racah type is given under the condition that Q is not a unit root.By using the representation theory of quantum envelope algebra Uso _ s _ 3, the classification of almost bipartite Leonard triples of q-Racah type. 3 is given under the condition that Q is not a unit root.Let K be an algebraic closed field characterized by zero.In this paper, we select a special Leonard pair of Bannai/Ito type and almost bipartite Leonard to form a Leonard triple. We construct all A ~ 蔚 鈭，

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