The Bannai-Ito algebra $BI(n)$ is viewed as the centralizer of the action of $\mathfrak{osp}(1|2)$ in the $n$-fold tensor product of the universal algebra of this Lie superalgebra. The generators of this centralizer are constructed with the help of the universal $R$-matrix of $\mathfrak{osp}(1|2)$. The specific structure of the $\mathfrak{osp}(1|2)$ embeddings to which the centralizing elements are attached as Casimir elements is explained. With the generators defined, the structure relations of $BI(n)$ are derived from those of $BI(3)$ by repeated action of the coproduct and using properties of the $R$-matrix and of the generators of the symmetric group $\mathfrak S_n$.

中文翻译：

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Bannai–Ito 代数和 $$\pmb {\mathfrak {osp}}(1|2)$$osp(1|2) 的通用 R 矩阵

Bannai-Ito 代数 $BI(n)$ 被视为 $\mathfrak{osp}(1|2)$ 在这个 Lie 超代数的通用代数的 $n$-fold 张量积中的作用的中心化器。这个中心化器的生成器是在 $\mathfrak{osp}(1|2)$ 的通用 $R$-矩阵的帮助下构建的。解释了 $\mathfrak{osp}(1|2)$ 嵌入的具体结构，中心化元素作为 Casimir 元素附加到这些嵌入中。定义了生成元后，$BI(n)$ 的结构关系通过余积的重复作用和使用 $R$-矩阵和对称群生成元的性质从 $BI(3)$ 的结构关系导出$\mathfrak S_n$。