论文标题
张量产品和固定操作员，用于Lie代数$ \ Mathfrak {SL}（2）（2）\ ltimes V（M）$
Tensor products and intertwining operators for uniserial representations of the Lie algebra $\mathfrak{sl}(2)\ltimes V(m)$
论文作者
论文摘要
Let $\mathfrak{g}_m=\mathfrak{sl}(2)\ltimes V(m)$, $m\ge 1$, where $V(m)$ is the irreducible $\mathfrak{sl}(2)$-module of dimension $m+1$ viewed as an abelian Lie algebra.众所周知，单性$ \ mathfrak {g} _m $ - 模型的同构类别由一个家庭组成，例如$ z $，包含任意构图长度的模块，以及一些具有组成长度的特殊模块$ \ le 4 $。 令$ v $和$ w $为两个单级$ \ mathfrak {g} _m $ - 类型$ z $的模块。在本文中，我们通过明确给出最高的权重向量来获得$ \ text {soc}（v \ otimes w）$的$ \ mathfrak {sl}（2）$ - 模块分解。事实证明，$ \ text {soc}（v \ otimes w）$是免费的。粗略地说，$ \ text {soc}（v \ otimes w）= \ text {soc}（v）（v）\ otimes \ otimes \ text {soc}（w）$在一半的情况下，在这些情况下，我们通过证明$ v \ otimes w $的完整socle系列，通过证明$ \ \ text {soc} {soc} {soc} { w）= \ sum_ {i = 0}^{t} \ text {soc}^{i+1}（v）（v）\ otimes \ text {soc}^{t+1-i}（t+1-i}（w）$ for $ t \ ge0 $。 作为这些结果的应用，我们获得了$ v $和$ w $，$ \ mathfrak {g} _m $ -Module同构的空间$ \ text {hom} _ {\ mathfrak {g} _m} _m} _m}（v，w）$不是零，在哪种情况下为1 dimensional。最后，我们证明，对于$ m \ ne 2 $，如果$ u $是两个单级$ \ mathfrak {g} _m $ $ -Modules type $ z $的张量产品，则这些因素由$ u $确定。我们提供了一个程序来确定$ u $的因素。
Let $\mathfrak{g}_m=\mathfrak{sl}(2)\ltimes V(m)$, $m\ge 1$, where $V(m)$ is the irreducible $\mathfrak{sl}(2)$-module of dimension $m+1$ viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial $\mathfrak{g}_m$-modules consist of a family, say of type $Z$, containing modules of arbitrary composition length, and some exceptional modules with composition length $\le 4$. Let $V$ and $W$ be two uniserial $\mathfrak{g}_m$-modules of type $Z$. In this paper we obtain the $\mathfrak{sl}(2)$-module decomposition of $\text{soc}(V\otimes W)$ by giving explicitly the highest weight vectors. It turns out that $\text{soc}(V\otimes W)$ is multiplicity free. Roughly speaking, $\text{soc}(V\otimes W)=\text{soc}(V)\otimes \text{soc}(W)$ in half of the cases, and in these cases we obtain the full socle series of $V\otimes W$ by proving that $ \text{soc}^{t+1}(V\otimes W)=\sum_{i=0}^{t} \text{soc}^{i+1}(V)\otimes \text{soc}^{t+1-i}(W)$ for all $t\ge0$. As applications of these results, we obtain for which $V$ and $W$, the space of $\mathfrak{g}_m$-module homomorphisms $\text{Hom}_{\mathfrak{g}_m}(V,W)$ is not zero, in which case is 1-dimensional. Finally we prove, for $m\ne 2$, that if $U$ is the tensor product of two uniserial $\mathfrak{g}_m$-modules of type $Z$, then the factors are determined by $U$. We provide a procedure to identify the factors from $U$.
