混合张量积与Capelli型行列式
Mixed tensor products and Capelli-type determinants
摘要：
本文研究了从泛包络代数$U=U(\mathfrak{gl}(n+1))$到微分算子代数$\mathcal D'(n)$与$U(\mathfrak{gl}(n))$的张量积的同态$ρ$的性质。我们发现一个公式的图像下$ρ$的$\mathfrak{gl}(n+1)$的Capelli行列式，特别是图像下的$ρ$的Gelfand发电机的中心$Z(\mathfrak{gl}(n+1))$的$U$。这个公式是证明有关$ρ$相应的哈里什-钱德拉同构，或者，通过使用一个纯粹的计算方法。此外，我们定义了一个同态从$\mathcal D'(n) \otimes U(\mathfrak{gl}(n))$到一个代数包含$U$作为一个子代数，使$σ(ρ(u)) - u \in G_1 U$，对所有的$u \in U$，其中$G_1 = \sum_{i=0}^{n} E_{ii}$。
英文摘要：
In this paper we study properties of a homomorphism $ρ$ from the universal enveloping algebra $U=U(\mathfrak{gl}(n+1))$ to a tensor product of an algebra $\mathcal D'(n)$ of differential operators and $U(\mathfrak{gl}(n))$. We find a formula for the image of the Capelli determinant of $\mathfrak{gl}(n+1)$ under $ρ$, and, in particular, of the images under $ρ$ of the Gelfand generators of the center $Z(\mathfrak{gl}(n+1))$ of $U$. This formula is proven by relating $ρ$ to the corresponding Harish-Chandra isomorphisms, and, alternatively, by using a purely computational approach. Furthermore, we define a homomorphism from $\mathcal D'(n) \otimes U(\mathfrak{gl}(n))$ to an algebra containing $U$ as a subalgebra, so that $σ(ρ(u)) - u \in G_1 U$, for all $u \in U$, where $G_1 = \sum_{i=0}^{n} E_{ii}$.
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