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arXiv 0802.1096math.RAmath.CO
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三对角对的结构

The structure of a tridiagonal pair

Kazumasa Nomura, Paul Terwilliger

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18 pages
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摘要：

设$K$表示域，$V$表示$K$上具有有限正维数的向量空间。我们考虑一对线性变换$A:V \to V$和$A^*:V \to V$，它们满足下列条件：（i）每个$A,A^*$都是可对角化的;（ii）存在$A$的特征空间的序$\{V_i\}_{i=0}^d$使得$A^* V_i \subseteq V_{i-1} + V_i + V_{i+1}$对于$0 \leq i \leq d$，其中$V_{-1}=0$和$V_{d+1}=0$ ;（iii）存在$A^*$的特征空间的序$\{V^*_i\}_{i=0}^δ$使得$A V^*_i \subseteq V^*_{i-1} + V^*_i + V^*_{i+1}$对于$0 \leq i \leq δ$，其中$V^*_{-1}=0$和$V^*_{δ+1}=0$ ;（iv）不存在$V$的子空间$W$使得$AW \subseteq W$，$A^* W \subseteq W$，$W \neq 0$，$W \neq V$。我们在$V$上称这样的对为三对角对。已知$d=δ$和$0 \leq i \leq d$的尺寸$V_i, V_{d-i}, V^*_i, V^*_{d-i}$一致。在本文中，我们证明了以下（i）-（iv）保持$K$是代数闭的：（i）每个$V_0$，$V^*_0$，$V_d$，$V^*_d$具有1维。(ii)存在$V$上的非退化对称双线性形式$(,)$，使得对所有的$u,v \in V$都有$(Au,v)=(u,Av)$和$(A^*u,v)=(u,A^*v)$. (iii)$End(V)$存在唯一的反自同构，它修复了每个$A,A^*$。(iv)对$A,A^*$由数据$(\{þ_i\}_{i=0}^d; \{þ^*_i\}_{i=0}^d; \{ζ_i\}_{i=0}^d)$确定直到同构，其中$þ_i$（resp. $þ^*_i$）是$A$（resp. $V_i$上的$A^*$（分别为$V^*_i$），并且$\{ζ_i\}_{i=0}^d$是$A,A^*$的分裂序列，对应于$\{þ_i\}_{i=0}^d$和$\{þ^*_i\}_{i=0}^d$。

英文摘要：

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\{V_i\}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_i + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\{V^*_i\}_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_i + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv)there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a tridiagonal pair on $V$. It is known that $d=δ$ and for $0 \leq i \leq d$ the dimensions of $V_i, V_{d-i}, V^*_i, V^*_{d-i}$ coincide. In this paper we show that the following (i)--(iv) hold provided that $K$ is algebraically closed: (i) Each of $V_0$, $V^*_0$, $V_d$, $V^*_d$ has dimension 1. (ii) There exists a nondegenerate symmetric bilinear form $(,)$ on $V$ such that $(Au,v)=(u,Av)$ and $(A^*u,v)=(u,A^*v)$ for all $u,v \in V$. (iii) There exists a unique anti-automorphism of $End(V)$ that fixes each of $A,A^*$. (iv) The pair $A,A^*$ is determined up to isomorphism by the data $(\{þ_i\}_{i=0}^d; \{þ^*_i\}_{i=0}^d; \{ζ_i\}_{i=0}^d)$, where $þ_i$ (resp. $þ^*_i$) is the eigenvalue of $A$ (resp. $A^*$) on $V_i$ (resp. $V^*_i$), and $\{ζ_i\}_{i=0}^d$ is the split sequence of $A,A^*$ corresponding to $\{þ_i\}_{i=0}^d$ and $\{þ^*_i\}_{i=0}^d$.

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