与$\Bbb{C}_{q}(osp(1,2))$相关的不变量和正交$q$ -多项式

Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra

New convolution identities for orthogonal polynomials belonging to the $q=-1$ analog of the Askey-scheme are obtained. A specialization of the Chihara polynomials will play a central role as the eigenfunctions of a special element of the Lie superalgebra $\mathfrak{osp}(1|2)$ in the positive discrete series representation. Using the Clebsch-Gordan coefficients, a convolution identity for the Specialized Chihara, the dual -1 Hahn and the Big -1 Jacobi polynomials is found. Using the Racah coefficients, a convolution identity for the Big -1 Jacobi and the Bannai-Ito polynomials is found. Finally…

量子超代数$U_q[osp(2n+1/2m)]$的Green生成元描述

A decription of the quantum superalgebra $U_q[osp(2n+1/2m)]$ via Green generators

利用一组新的生成元（称为Green生成元）描述了正交辛李超代数$osp(2n+1/2m)$及其变形的类似物$U_q[osp(2n+1/2m)]$.这些发电机非常不同于著名的Chevalley发电机。在数学上，格林生成元是根向量，对应于正正交根和负正交根。物理上，它们由（变形的）准玻色算符的$m$对和（变形的）准费米算符的$n$对组成。

泛T-矩阵、OSp_q（1/2）和小Q-雅可比多项式的表示

Universal T-matrix, Representations of OSp_q(1/2) and Little Q-Jacobi Polynomials

我们得到了包含量子超代数U_q[osp（1/2）]和相应的超群OSp_q（1/2）对偶的泛T矩阵的封闭形式表达式。这个普适T-矩阵的经典q->1极限产生了未变形的OSp（1/2）超群的群元。利用所述泛T矩阵和已知的对偶相关变形超代数U_q[osp（1/2）]的有限维表示，很容易构造出量子超群OSp_q（1/2）的有限维表示.进一步导出了量子超群OSp_q（1/2）的乘积律、递推关系和表示的正交性。证明了这些表示矩阵的元素都表示为Q = -q的小Q-雅可比多项式。给出了泛T-矩阵在泛R-矩阵上的两个互补奇异映射。

The quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ and a $q$-generalization of the Bannai-Ito polynomials

The Racah problem for the quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ is considered. The intermediate Casimir operators are shown to realize a $q$-deformation of the Bannai-Ito algebra. The Racah coefficients of $\mathfrak{osp}_q(1|2)$ are calculated explicitly in terms of basic orthogonal polynomials that $q$-generalize the Bannai-Ito polynomials. The relation between these $q$-deformed Bannai-Ito polynomials and the $q$-Racah/Askey-Wilson polynomials is discussed.

Yang-Baxter R-operators for osp superalgebras

We study Yang-Baxter equations with orthosymplectic supersymmetry. We extend a new approach of the construction of the spinor and metaplectic $\hat{\cal R}$-operators with orthogonal and symplectic symmetries to the supersymmetric case of orthosymplectic symmetry. In this approach the orthosymplectic $\hat{\cal R}$-operator is given by the ratio of two operator valued Euler Gamma-functions. We illustrate this approach by calculating such $\hat{\cal R}$ operators in explicit form for special cases of the $osp(n|2m)$ algebra, in particular for a few low-rank cases. We also propose a novel, simpl…

Quantum Shuffles and Quantum Supergroups of Basic Type

We initiate the study of several distinguished bases for the positive half of a quantum supergroup $U_q$ associated to a general super Cartan datum $(\mathrm{I}, (\cdot,\cdot))$ of basic type inside a quantum shuffle superalgebra. The combinatorics of words for an arbitrary total ordering on $\mathrm{I}$ is developed in connection with the root system associated to $\mathrm{I}$. The monomial, Lyndon, and PBW bases of $U_q$ are constructed, and moreover, a direct proof of the orthogonality of the PBW basis is provided within the framework of quantum shuffles. Consequently, the canonical basis i…