论文标题
沙波瓦洛夫元素的分解
Factorization of Shapovalov elements
论文作者
论文摘要
Shapovalov元素$θ_{β，m} $是由正块$β$和正整数$ m $的经典或量子通用包围代数的Borel子代数中的特殊元素。它们将可还原的Verma模块的规范发电机与其Verma子模块的最高矢量相关联。对于$ m = 1 $，可以明确地将它们作为逆Shapovalov表格的矩阵元素获得。我们将这种方法扩展到所有$β$的$ m> 1 $，但在$ \ mathfrak {g} _2 $，$ \ mathfrak {f} _4 $和$ \ mathfrak {e} _8 $，呈现$θ_{β，m} $ a作为量子$ $ $ $β$的产品的产物。
Shapovalov elements $θ_{β,m}$ are special elements in a Borel subalgebra of a classical or quantum universal enveloping algebra parameterized by a positive root $β$ and a positive integer $m$. They relate the canonical generator of a reducible Verma module with highest vectors of its Verma submodules. For $m=1$, they can be explicitly obtained as matrix elements of the inverse Shapovalov form. We extend this approach to $m>1$ for all $β$ but three roots in $\mathfrak{g}_2$, $\mathfrak{f}_4$, and $\mathfrak{e}_8$, presenting $θ_{β,m}$ as a product of matrix elements of weight $β$.
