$sl_2$的量子环代数与$q$-拉卡型二元函数
The quantum loop algebra of $sl_2$ and $q$-Racah type bivariate functions
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中文总结 AI 辅助
该研究借助$sl_2$的量子环代数$\mathcal{L} U_q sl_2$的表示理论，构建统一代数框架处理两个六参数二元$q$-拉卡型函数族。通过特定元素得到相关重叠系数，在一定条件下这些元素对呈现特定类型，还讨论了系数性质及与其他代数的关系。
中文摘要 AI 辅助
利用$sl_2$的量子环代数$\mathcal{L} U_q sl_2$的表示理论，给出了两个不同的六参数二元$q$-拉卡型函数族的统一代数框架。分析从$\mathcal{L} U_q sl_2$的两个左右余理想子代数和六个交换子代数开始，由$\mathcal{L} U_q sl_2 \otimes \mathcal{L} U_q sl_2$中的八个元素构建。这八个元素依赖于两个标量$a,b \in {\mathbb C}^*$，并在有限维向量空间上对角化。二元$q$-拉卡型函数被解释为与由评估参数$u_1,u_2$标记的$\mathcal{L} U_q sl_2$的张量积（评估）表示的六个“特殊”特征基相关的重叠系数。在对$a,b,u_1,u_2$的某些条件下，表明元素对的一个子集作为I型三对角对（也称为$q$-拉卡型）起作用。对于$u_1/u_2 = 1$，推测元素对的另一个子集作为因式分解的伦纳德对起作用。从而得到了与不同对相关的各种特征基的相应重叠系数。还讨论了它们的一些性质，以及它们与已知的特拉特尼克型二元多项式和秩2阿斯凯 - 威尔逊代数的关系。
英文摘要
A unified algebraic framework for two different six-parameter families of bivariate $q$-Racah type functions is given using the representation theory of the quantum loop algebra $\mathcal{L} U_q sl_2$ of $sl_2$. The starting point of the analysis is two left and right coideal subalgebras of $\mathcal{L} U_q sl_2$ and six commutative subalgebras, built from eight elements in $\mathcal{L} U_q sl_2 \otimes \mathcal{L} U_q sl_2$. The eight elements depend on two scalars $a,b \in {\mathbb C}^*$, and are diagonalized on a finite-dimensional vector space. The bivariate $q$-Racah type functions are interpreted as the overlap coefficients relating six `distinguished' eigenbases parametrized by $a,b$ of the tensor product (evaluation) representations of $\mathcal{L} U_q sl_2$ labeled by the evaluation parameters $u_1,u_2$. Upon certain conditions on $a,b,u_1,u_2$, it is shown that a subset of pairs of elements act as tridiagonal pairs of type I (also called $q$-Racah type). For $u_1/u_2=1$, another subset of pairs of elements are conjectured to act as factorized Leonard pairs. Thus, in both cases corresponding overlap coefficients relating the various eigenbases associated with different pairs are obtained. Some of their properties are also discussed, as well as their relation with known bivariate polynomials of Tratnik type and the rank 2 Askey--Wilson algebra.
