论文标题
$ Q $ -ONSAGER代数$ \ Mathcal O_Q $的扭曲有限维模块通过Lusztig自动形态
Twisting finite-dimensional modules for the $q$-Onsager algebra $\mathcal O_q$ via the Lusztig automorphism
论文作者
论文摘要
$ q $ -onsager代数$ \ MATHCAL O_Q $由两个发电机$ a $，$ a^*$和两个关系（称为$ q $ -dolan/grady Relative）定义。最近，P。Baseilhac和S. Kolb发现了$ \ Mathcal o_q $的自动形态$ l $ l $，它可以修复$ a $，并将$ a^*$发送到$ a^*$，$ a^*$，$ a^2a^*$，$ aa^*$，$ aa^*a^*a $，$ a $，$ a^*a^*a^*a^2 $的线性组合。令$ v $表示有限尺寸的不可约$ \ Mathcal O_Q $ - $ a $ a $，$ a^*$的每个尺寸都可以对角线。众所周知，$ a $，$ a^*$在$ v $上作为$ q $ -racah类型的$ v $ ACT，可访问四个熟悉的元素$ k $，$ b $，$ k^\ downarrow $，$ b^\ downarrow $，$ b^\ downarrow $ in $ {\ rm end end} $ in $ { 我们在{\ rm end}（v）$中显示一个可逆的$ h \，以便$ l（x）= h^{ - 1} x h $ on $ v $ in $ x \ in \ in \ mathcal o_q $ in $ x \ in $ v $。我们描述当$ k $，$ b $，$ k^\ downarrow $，$ b^\ downarrow $中的一个中会发生什么。例如，$ h^{ - 1} kh = a^{ - 1} a-a^{ - 2} k^{ - 1} $，其中$ a $是一种用来描述$ a $ a $ on $ v $的特定标量。我们使用共轭结果来比较$ v $上的$ a $，$ a^*$，$ a^*$，$ a^*$，$ a^*$，$ a^*$，$ a^*）$的eigenspace分解。在此比较中，我们使用了公平三倍的概念。这是$ {\ rm end}（v）$中元素的3键，因此任何两个都满足$ q $ -weyl关系。我们的比较涉及八个公平三元组。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 $出现在S. Bockting-Conrad的早期工作中，这些工作是关于三agonal对的双重降低操作员$ψ$。
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.
