Irreducible Witt modules from Weyl modules and $\mathfrak{gl}_{n}$-modules

TL;DR

This paper constructs a broad class of irreducible modules over Witt algebras from Weyl and rfgl_n modules, providing criteria for irreducibility and classifying submodules.

Contribution

It introduces a method to generate irreducible Witt algebra modules from Weyl and rfgl_n modules using Shenb4s monomorphism, expanding the known family of modules.

Findings

Established necessary and sufficient conditions for irreducibility.

Classified all submodules when reducible.

Constructed many new irreducible weight and non-weight modules.

Abstract

For an irreducible module $P$ over the Weyl algebra $\mathcal{K}_n^+$ (resp. $\mathcal{K}_n$) and an irreducible module $M$ over the general liner Lie algebra $\mathfrak{gl}_n$, using Shen's monomorphism, we make $P\otimes M$ into a module over the Witt algebra $W_n^+$ (resp. over $W_n$). We obtain the necessary and sufficient conditions for $P\otimes M$ to be an irreducible module over $W_n^+$ (resp. $W_n$), and determine all submodules of $P\otimes M$ when it is reducible. Thus we have constructed a large family of irreducible weight modules with many different weight supports and many irreducible non-weight modules over $W_n^+$ and $W_n$.