$q$ -Serre型三对角对及其线性扰动
Tridiagonal pairs of $q$-Serre type and their linear perturbations
摘要：
三对角对是有限维非零向量空间上可对角化的线性映射的有序对，每个映射以块三对角方式作用于另一个的特征空间。我们考虑一个$q$ -Serre型的三对角对$(A, A^*)$;对于这样一个对，映射$A$和$A^*$满足$q$ -Serre关系.文献中有一个线性映射$K$，用于描述$A$和$A^*$之间的关系。研究了一对线性映射$B=A$和$B^* = tA^* + (1-t)K$，其中$t$是任意标量.我们的目标是找到一个必要和充分条件$t$对$(B, B^*)$是一个三对角对。我们表明，$(B, B^*)$是一个三对角对当且仅当$t \neq 0$和$P \bigl( t(q-q^{-1})^{-2} \bigr)\not=0$，其中$P$是一个特定的多项式附加到$(A, A^*)$称为Drinfel的多项式。
英文摘要：
A tridiagonal pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on the eigenspaces of the other in a block-tridiagonal fashion. We consider a tridiagonal pair $(A, A^*)$ of $q$-Serre type; for such a pair the maps $A$ and $A^*$ satisfy the $q$-Serre relations. There is a linear map $K$ in the literature that is used to describe how $A$ and $A^*$ are related. We investigate a pair of linear maps $B=A$ and $B^* = tA^* + (1-t)K$, where $t$ is any scalar. Our goal is to find a necessary and sufficient condition on $t$ for the pair $(B, B^*)$ to be a tridiagonal pair. We show that $(B, B^*)$ is a tridiagonal pair if and only if $t \neq 0$ and $P \bigl( t(q-q^{-1})^{-2} \bigr)\not=0$, where $P$ is a certain polynomial attached to $(A, A^*)$ called the Drinfel'd polynomial.
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