Exponentiation and Fourier transform of tensor modules of $\mathfrak{sl} (n+1)$

TL;DR

This paper introduces a new class of modules for rak{sl}(n+1) using exponentiation and Fourier transform, expanding the understanding of weight modules and their structures.

Contribution

It constructs and analyzes a novel class of tensor modules for rak{sl}(n+1) via explicit differential operator presentations, including isomorphism and simplicity criteria.

Findings

Defined modules T(g,V,S) with explicit differential operator representations

Established isomorphism and simplicity criteria for these modules

Generated new tensor coherent families of rak{sl}(n+1)

Abstract

With the aid of the exponentiation functor and Fourier transform we introduce a class of modules $T(g,V,S)$ of $\mathfrak{sl} (n+1)$ of mixed tensor type. By varying the polynomial $g$, the $\mathfrak{gl}(n)$-module $V$, and the set $S$, we obtain important classes of weight modules over the Cartan subalgebra $\mathfrak h$ of $\mathfrak{sl} (n+1)$, and modules that are free over $\mathfrak h$. Furthermore, these modules are obtained through explicit presentation of the elements of $\mathfrak{sl} (n+1)$ in terms of differential operators and lead to new tensor coherent families of $\mathfrak{sl} (n+1)$. An isomorphism theorem and simplicity criterion for $T(g,V,S)$ is provided.