Uq[osp（m）]的Casimir不变量的特征值|n）]

Bannai-Ito algebras and the $osp(1,2)$ superalgebra

The Bannai-Ito algebra $B(n)$ of rank $(n-2)$ is defined as the algebra generated by the Casimir operators arising in the $n$-fold tensor product of the $osp(1,2)$ superalgebra. The structure relations are presented and representations in bases determined by maximal Abelian subalgebras are discussed. Comments on realizations as symmetry algebras of physical models are offered.

量子正交辛超代数的杨-巴克斯特方程解与Casimir不变量

Solutions to the Yang-Baxter Equation and Casimir Invariants for the Quantised Orthosymplectic Superalgebra

在过去的15年里，量子超代数已经被用来模拟超对称量子系统。一类拟三角Hopf超代数，它们都包含一个泛$R$ -矩阵，该矩阵自动满足杨-巴克斯特方程.将向量表示应用于通用$R$ -矩阵的左手侧给出Lax算子。这些在数学物理中具有重要意义，因为它们以任意表示形式提供了杨-巴克斯特方程的解，从而产生了可积模型。本文对量子正交辛超代数$U_q[osp(m|n)]$构造了一个Lax算子，它对所有$m > 2, n \geq 2$都成立，其中$n$是偶数.然后，这可以被用来找到一个解决方案的杨-巴克斯特方程在任意表示的$U_q[osp(m|n)]$，与矢量表示的例子中给出的细节。在研究由杨-巴克斯特方程的解所产生的可积模型时，需要理解超代数的表示论。找到系统的Casimir不变量并探索它们的行为有助于这种理解。本文利用Lax算子构造了$U_q[osp(m|n)]$的无限个Casimir不变量族，并计算了它们在任意不可约表示中的特征值。

The quantum Casimir operators of $\Uq$ and their eigenvalues

We show that the quantum Casimir operators of the quantum linear group constructed in early work of Bracken, Gould and Zhang together with one extra central element generate the entire center of $\Uq$. As a by product of the proof, we obtain intriguing new formulae for eigenvalues of these quantum Casimir operators, which are expressed in terms of the characters of a class of finite dimensional irreducible representations of the classical general linear algebra.

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.

论卡西米尔的幽灵

On Casimir's Ghost

定义了OSP（1）的泛包络超代数|2n）一个非标准伴随作用，赋予它模结构。这使得，特别是，构建一个玻色子运营商的反交换与所有的费米子发电机，这似乎是平方根的某个卡西米尔运营商。

q-代数U '_q（so_n）的Casimir元及其特征值

On the Casimir Elements of q-Algebras U'_q(so_n) and Their Eigenvalues in Representations

非标准q-变形代数U '_q（so_n）具有Gel'fand-Tsetlin型表示的q-模拟.对于这些q-代数，求出了所有的Casimir元（对应于so（n）的Casimir元的基集），并明确给出了它们在不可约表示内的特征值.