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- W3035084026 abstract "For any $a,binmathbb C$, $W(a,b)$ is the Lie algebra with basis ${L_m,M_m,|,minmathbb Z}$ 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 $m,ninmathbb Z$. For any $lambdainmathbb C^*,$ $alphainmathbb C$, $h:=h(t)inmathbb C[t]$, 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 $U(mathbb C L_0oplusmathbb C W_0)$ of $mathbb C L_0oplusmathbb C W_0$. 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 $b=-1$, ${rm deg}, h(t)=1$, $alphaneq0$. 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 $lambda_i,alpha_iinmathbb C^*,$ $h_iinmathbb C[t]$ 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 $mgeq R_V$." @default.
- W3035084026 created "2020-06-19" @default.
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- W3035084026 date "2017-08-24" @default.
- W3035084026 modified "2023-09-27" @default.
- W3035084026 title "Submodule structures of $mathbb C[s,t]$ over $W(0,b)$ and a new class of irreducible modules over the Virasoro algebra." @default.
- W3035084026 hasPublicationYear "2017" @default.
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