论文标题
无限尺寸简单$ \ mathfrak {osp}（1 | 2n）$ - 尊重分支$ \ mathfrak {osp}（1 | 2n）\ supset \ supset \ mathfrak {gl}（n）$的模块
Bases for infinite dimensional simple $\mathfrak{osp}(1|2n)$-modules respecting the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$
论文作者
论文摘要
我们研究了分支$ \ mathfrak {osp}（1 | 2n）\ supset \ mathfrak {gl}（n）$对特定类别的简单无限尺寸$ \ mathfrak {osp}（osp}（1 | 2n）$ - modules $ l（p）$ l（p）$。在第一部分中，我们使用诸如Young Tableaux和Young子组之类的组合方法来为$ L（p）$构建一个尊重该分支的新基础，并且我们以两种不同的方式明确表达了基础元素。首先是$ \ mathfrak {gl}（n）$的负根矢量的单元，作用在$ l（p）$中的$ \ mathfrak {gl}（n）$ - 最高重量向量，然后在$ \ m m iosp {osp}（osp}（1 | 2n）$ $ \ n $ $ \ math的发电机中，在$ l（p）$中| $ l（p）$中的重量向量。在第二部分中，我们使用极值投影仪和Mickelsson-Zhelobenko代数的理论，为与分支$ \ Mathfrak {osp}（1 | 2n）\ supset \ supset \ mathfrak {gl}（gl}（gl}（n）$。我们使用饲养的操作员为$ l（p）$的Gel'fand-Zetlin基础的元素提供新的表达方式，作为$ u（\ Mathfrak {osp}（1 | 2n））$的运算符，以$ \ \ \ \ \ \ \ \\ Mathfrak {osp}（osp}（1 | 2n）$ - 最低级$ - 最低量Vector $（p）$（p）。我们观察到，$ l（p）$的凝胶fand-zetlin基础与三角形过渡矩阵早期构建的基础有关。我们以一个详细的示例结束了论文，以处理案例$ n = 3 $。
We study the effects of the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$ on a particular class of simple infinite-dimensional $\mathfrak{osp}(1|2n)$-modules $L(p)$ characterized by a positive integer $p$. In the first part we use combinatorial methods such as Young tableaux and Young subgroups to construct a new basis for $L(p)$ that respects this branching and we express the basis elements explicitly in two distinct ways. First as monomials of negative root vectors of $\mathfrak{gl}(n)$ acting on the $\mathfrak{gl}(n)$-highest weight vectors in $L(p)$ and then as polynomials in the generators of $\mathfrak{osp}(1|2n)$ acting on the $\mathfrak{osp}(1|2n)$-lowest weight vector in $L(p)$. In the second part we use extremal projectors and the theory of Mickelsson-Zhelobenko algebras to give new explicit constructions of raising and lowering operators related to the branching $\mathfrak{osp}(1|2n)\supset \mathfrak{gl}(n)$. We use the raising operators to give new expressions for the elements of the Gel'fand-Zetlin basis for $L(p)$ as monomials of operators from $U(\mathfrak{osp}(1|2n))$ acting on the $\mathfrak{osp}(1|2n)$-lowest weight vector in $L(p)$. We observe that the Gel'fand-Zetlin basis for $L(p)$ is related to the basis constructed earlier in the paper by a triangular transition matrix. We end the paper with a detailed example treating the case $n=3$.
