Classification of finite irreducible conformal modules over Lie conformal algebra <inline-formula><tex-math id="M1">$ \mathcal{W}(a, b, r) $</tex-math></inline-formula>

Abstract

&lt;p style='text-indent:20px;'&gt;We study a family of non-simple Lie conformal algebras &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;$ \mathcal{W}(a,b,r) $&lt;/tex-math&gt;&lt;/inline-formula&gt; (&lt;inline-formula&gt;&lt;tex-math id="M3"&gt;$ a,b,r\in {\mathbb{C}} $&lt;/tex-math&gt;&lt;/inline-formula&gt;) of rank three with free &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;$ {\mathbb{C}}[{\partial}] $&lt;/tex-math&gt;&lt;/inline-formula&gt;-basis &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;$ \{L, W,Y\} $&lt;/tex-math&gt;&lt;/inline-formula&gt; and relations &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;$ [L_{\lambda} L] = ({\partial}+2{\lambda})L,\ [L_{\lambda} W] = ({\partial}+ a{\lambda} +b)W,\ [L_{\lambda} Y] = ({\partial}+{\lambda})Y,\ [Y_{\lambda} W] = rW $&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;$ [Y_{\lambda} Y] = [W_{\lambda} W] = 0 $&lt;/tex-math&gt;&lt;/inline-formula&gt;. In this paper, we investigate the irreducibility of all free nontrivial &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;$ \mathcal{W}(a,b,r) $&lt;/tex-math&gt;&lt;/inline-formula&gt;-modules of rank one over &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;$ {\mathbb{C}}[{\partial}] $&lt;/tex-math&gt;&lt;/inline-formula&gt; and classify all finite irreducible conformal modules over &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;$ \mathcal{W}(a,b,r) $&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;