Mixed Tensor Products, Capelli Berezinians, and Newton's Formula for $\mathfrak{gl}(m|n)$

TL;DR

This work extends mixed tensor (tensor) module theory and Capelli-type identities from the classical Lie algebra setting to the Lie superalgebra $\mathfrak{gl}(m+1|n)$. The authors construct a central-element–indexed family of superalgebra homomorphisms $\varphi_R$ from $U(\mathfrak{gl}(m+1|n))$ to $\mathcal{D}'(m|n) \otimes U(\mathfrak{gl}(m|n))$, relate them to Harish-Chandra homomorphisms, and prove a super Newton's formula connecting CapelliBerezinians and Gelfand generators. They develop inflation mechanisms for representations, identify when inflations are simple, and compute the images of central elements under these maps, including an explicit description of the images of Gelfand invariants. A key result is that the kernel of the distinguished map $\varphi_{R_1}$ is generated by the first Gelfand invariant $G_1^{\mathfrak{gl}(m+1|n)}$, providing a clear center-to-center bridge in the super setting. Collectively, these results deepen the understanding of central structure, representations, and Capelli-type identities in the superalgebra context and lay groundwork for potential geometric interpretations and further structural developments.

Abstract

In this paper, we extend the results of Grantcharov and Robitaille in 2021 on mixed tensor products and Capelli determinants to the superalgebra setting. Specifically, we construct a family of superalgebra homomorphisms $\varphi_R : U(\mathfrak{gl}(m+1|n)) \rightarrow \mathcal{D}'(m|n) \otimes U(\mathfrak{gl}(m|n))$ for a certain space of differential operators $\mathcal{D}'(m|n)$ indexed by a central element $R$ of $\mathcal{D}'(m|n) \otimes U(\mathfrak{gl}(m|n))$. We then use this homomorphism to determine the image of Gelfand generators of the center of $U(\mathfrak{gl}(m+1|n))$. We achieve this by first relating $\varphi_R$ to the corresponding Harish-Chandra homomorphisms and then proving a super-analog of Newton's formula for $\mathfrak{gl}(m)$ relating Capelli generators and Gelfand generators. We also use the homomorphism $\varphi_R$ to obtain representations of $U(\mathfrak{gl}(m+1|n))$ from those of $U(\mathfrak{gl}(m|n))$, and find conditions under which these inflations are simple. Finally, we show that for a distinguished central element $R_1$ in $\mathcal{D}'(m|n)\otimes U(\mathfrak{gl}(m|n))$, the kernel of $\varphi_{R_1}$ is the ideal of $U(\mathfrak{gl}(m+1|n))$ generated by the first Gelfand invariant $G_1$.

Theorems & Definitions (32)

• Lemma 2.3.1
• Theorem 3.1
• Corollary 3.2
• Proposition 3.3
• Proposition 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof