利用Lusztig自同构构造$q$代数$\mathcal O_q$的扭转有限维模
Twisting finite-dimensional modules for the $q$-Onsager algebra $\mathcal O_q$ via the Lusztig automorphism
摘要：
$q$-Onsager代数$\mathcal O_q$由两个生成元$A$、$A^*$和两个关系（称为$q$-Dolan/Grady关系）定义.最近P. Baseilhac和S. Kolb发现了$\mathcal O_q$的自同构$L$，它修复了$A$，并将$A^*$发送到$A^*$，$A^2A^*$，$AA^*A$，$A^*A^2$的线性组合。设$V$表示有限维至少为2的不可约$\mathcal O_q$ -模，在该模上$A$，$A^*$都是可对角化的.已知$A$、$A^*$作为$q$-Racah类型的三对角对作用于$V$，从而可以访问${\rm End}(V)$中的四个熟悉的元素$K$、$B$、$K^\downarrow$、$B^\downarrow$，这些元素用于比较$A$、$A^*$在$V$上的本征空间分解。我们显示一个可逆的$H \in {\rm End}(V)$，使得对于所有$X \in \mathcal O_q$，$L(X)=H^{-1} X H$在$V$上。我们描述了当$K$、$B$、$K^\downarrow$、$B^\downarrow$之一与$H$结合时发生的情况。例如，$H^{-1}KH=a^{-1}A-a^{-2}K^{-1}$，其中$a$是用于描述$V$上的$A$的特征值的特定标量。我们使用共轭的结果来比较$A$，$A^*$，$L^{\pm 1}(A^*)$在$V$上的特征空间分解。在这个比较中，我们使用公平三元组的概念;这是${\rm End}(V)$中的元素的3元组，使得任何两个满足$q$-Weyl关系。我们的比较涉及八个公平的三元组。其中之一是$a A - a^2 K$，$M^{-1}$，$K$其中$M= (a K-a^{-1} B)(a-a^{-1})^{-1}$ .地图$M$出现在早期的工作S。Bockting-Conrad关于三对角对的双降算子$ψ$的结果。
英文摘要：
The $q$-Onsager algebra $\mathcal O_q$ is defined by two generators $A$, $A^*$ and two relations, called the $q$-Dolan/Grady relations. Recently P. Baseilhac and S. Kolb found an automorphism $L$ of $\mathcal O_q$, that fixes $A$ and sends $A^*$ to a linear combination of $A^*$, $A^2A^*$, $AA^*A$, $A^*A^2$. Let $V$ denote an irreducible $\mathcal O_q$-module of finite dimension at least two, on which each of $A$, $A^*$ is diagonalizable. It is known that $A$, $A^*$ act on $V$ as a tridiagonal pair of $q$-Racah type, giving access to four familiar elements $K$, $B$, $K^\downarrow$, $B^\downarrow$ in ${\rm End}(V)$ that are used to compare the eigenspace decompositions for $A$, $A^*$ on $V$. We display an invertible $H \in {\rm End}(V)$ such that $L(X)=H^{-1} X H$ on $V$ for all $X \in \mathcal O_q$. We describe what happens when one of $K$, $B$, $K^\downarrow$, $B^\downarrow$ is conjugated by $H$. For example $H^{-1}KH=a^{-1}A-a^{-2}K^{-1}$ where $a$ is a certain scalar that is used to describe the eigenvalues of $A$ on $V$. We use the conjugation results to compare the eigenspace decompositions for $A$, $A^*$, $L^{\pm 1}(A^*)$ on $V$. In this comparison we use the notion of an equitable triple; this is a 3-tuple of elements in ${\rm End}(V)$ such that any two satisfy a $q$-Weyl relation. Our comparison involves eight equitable triples. One of them is $a A - a^2 K$, $M^{-1}$, $K$ where $M= (a K-a^{-1} B)(a-a^{-1})^{-1}$. The map $M$ appears in earlier work of S. Bockting-Conrad concerning the double lowering operator $ψ$ of a tridiagonal pair.
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