Simple $\mathcal{W}\ltimes\widehat{H_4}$-modules from tensor products

TL;DR

This paper classifies simple modules over the semi-direct product algebra involving the Witt algebra and loop Diamond algebra, constructing new modules from Weyl algebra modules and analyzing their simplicity and isomorphism classes.

Contribution

It introduces a method to construct and classify simple modules of the algebra using tensor products and provides criteria for their simplicity and isomorphism classes.

Findings

Constructed a family of simple modules from Weyl algebra modules.

Classified simple modules that are free of rank 1 over a certain subalgebra.

Established criteria for simplicity and determined isomorphism classes.

Abstract

This paper investigates simple modules of the semi-direct product algebra $\mathcal{W}\ltimes\widehat{H_4}$, where $\mathcal{W}$ is the Witt algebra and $\widehat{H_4}$ is the loop Diamond algebra. We first use simple modules over the Weyl algebra to construct a family of simple $\mathcal{W}\ltimes\widehat{H_4}$-modules. Then, we classify simple $\mathcal{W}\ltimes\widehat{H_4}$-modules that are free $U(\mathbb{C}L_0\oplus\mathbb{C} a_0)$-modules of rank 1. Finally, we give a necessary and sufficient condition for finitely many simple $U(\mathbb{C}L_0\oplus\mathbb{C}a_0)$-free modules to be simple, and then determine their isomorphism classes.