AI 中文总结
该研究针对二阶Racah代数，通过在三角形的二阶Jacobi代数微分模型中，利用三对角化将乘法算子提升为Racah型生成元，构造出其到二阶Jacobi代数的显式嵌入，确定了相关共同本征函数与重叠系数，还明确了二阶Jacobi代数作为二阶Racah代数收缩的生成机制。
AI 中文摘要
我们构造了二阶Racah代数$\boldsymbol{\frak{R}}_2$到二阶Jacobi代数$\boldsymbol{\frak{J}}_2$的显式嵌入。在三角形上$\boldsymbol{\frak{J}}_2$的微分模型中，我们回顾到$\boldsymbol{\frak{J}}_2$的子代数结构由五边形组织，其四条上边承载一阶Jacobi代数，底边承载一阶Racah代数。通过三对角化将两个乘法算子提升为Racah型生成元，可将五边形的每条边转化为一阶Racah代数，从而在$\boldsymbol{\frak{J}}_2$内部实现$\boldsymbol{\frak{R}}_2$。我们明确确定了不同可交换对的共同本征函数，其形式为高斯超几何函数与Jacobi多项式的乘积。计算了不同的重叠系数，涉及单变量Wilson多项式，以及对于一对基，涉及Tratnik型双变量Racah多项式。我们还回顾了$\boldsymbol{\frak{J}}_2$如何作为二阶Racah代数$\boldsymbol{\frak{R}}_2$的收缩而产生。
英文摘要
We construct an explicit embedding of the rank two Racah algebra $\mathfrak{R}_2$ into the rank two Jacobi algebra $\mathfrak{J}_2$. Working in the differential model of $\mathfrak{J}_2$ on the triangle, we recall that the subalgebra structure of $\mathfrak{J}_2$ is organized by a pentagon whose four upper edges carry rank one Jacobi algebras and whose base carries a rank one Racah algebra. Promoting the two multiplication operators to Racah type generators by tridiagonalization turns every edge of the pentagon into a rank one Racah algebra and realizes $\mathfrak{R}_2$ inside $\mathfrak{J}_2$. We determine the common eigenfunctions of different commuting pairs explicitly as products of Gauss hypergeometric functions and Jacobi polynomials. Different overlap coefficients are computed involving univariate Wilson polynomials as well as, for one pair of bases, bivariate Racah polynomials of Tratnik type. We also recall how $\mathfrak{J}_2$ arises as a contraction of the rank two Racah algebra $\mathfrak{R}_2$.
