Submodule structures of $\mathbb C[s,t]$ over $W(0,b)$ and a new class of irreducible modules over the Virasoro algebra

TL;DR

This paper classifies submodules of polynomial modules over certain Lie algebras, introduces new irreducible modules over the Virasoro algebra, and characterizes their properties and conditions for irreducibility.

Contribution

It determines all submodules of polynomial modules over $W(0,b)$, introduces a new class of irreducible Virasoro modules, and constructs a broad family of such modules via tensor products.

Findings

Submodules of $C[s,t]$ are fully characterized.

Irreducibility of modules $Phi(lambda,alpha,h)$ is established under specific conditions.

A large family of new irreducible Virasoro modules is constructed.

Abstract

For any $a,b\in\mathbb C$, $W(a,b)$ is the Lie algebra with basis $\{L_m,M_m\,|\,m\in\mathbb Z\}$ and relations $[L_m,L_n]=(n-m)L_{m+n},$ $[L_m,W_n]=(a+n+bm)W_{m+n}$, $[W_m,W_n]=0$ for $m,n\in\mathbb Z$. For any $\lambda\in\mathbb C^*,$ $\alpha\in\mathbb C$, $h:=h(t)\in\mathbb C[t]$, there exists a non-weight module over $W(0,b)$ (resp., $W(0,1)$), denoted by $\Phi(\lambda,\alpha,h)$ (resp. $\Theta(\lambda,h)$), which is defined on the space $\mathbb C[s,t]$ of polynomials on variables $s,t$ and is free of rank one over the enveloping algebra $U(\mathbb C L_0\oplus\mathbb C W_0)$ of $\mathbb C L_0\oplus\mathbb C W_0$. In the present paper, by introducing two sequences of useful operators on $\mathbb C[s,t]$, we determine all submodules of $\mathbb C[s,t]$. We also study submodules of $\mathbb C[s,t]$ regarded as modules over the Virasoro algebra $\mathscr V\!$ (with the trivial action of the center), and prove that these submodules are finitely generated if and only if ${\rm deg}\,h(t)\geq1$. In addition, it is proven that $\Phi(\lambda, \alpha,h)$ is an irreducible $\mathscr V\!$-module if and only if $b=-1$, ${\rm deg}\, h(t)=1$, $\alpha\neq0$. Finally, we obtain a large family of new irreducible modules over the Virasoro algebra $\mathscr V\!$, by taking various tensor products of a finite number of irreducible modules $\Phi(\lambda_i,\alpha_i, h_i)$ for $\lambda_i,\alpha_i\in\mathbb C^*,$ $h_i\in\mathbb C[t]$ with an irreducible $\mathscr V\!$-module $V$, where $V$ satisfies that there exists a nonnegative integer $R_V$ such that $L_m$ acts locally finitely on $V$ for $m\geq R_V$.