American Institute of Mathematical Sciences

We investigate the solvability of the matrix equation \begin{document}$ AX^{2} = B $\end{document} in which the multiplication is the semi-tensor product. Then compatible conditions on the matrices \begin{document}$ A $\end{document} and \begin{document}$ B $\end{document} are established in each case and necessary and sufficient condition for the solvability is discussed. In addition, concrete methods of solving the equation are provided.

We study a family of non-simple Lie conformal algebras \begin{document}$ \mathcal{W}(a,b,r) $\end{document} (\begin{document}$ a,b,r\in {\mathbb{C}} $\end{document}) of rank three with free \begin{document}$ {\mathbb{C}}[{\partial}] $\end{document}-basis \begin{document}$ \{L, W,Y\} $\end{document} and relations \begin{document}$ [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 $\end{document} and \begin{document}$ [Y_{\lambda} Y] = [W_{\lambda} W] = 0 $\end{document}. In this paper, we investigate the irreducibility of all free nontrivial \begin{document}$ \mathcal{W}(a,b,r) $\end{document}-modules of rank one over \begin{document}$ {\mathbb{C}}[{\partial}] $\end{document} and classify all finite irreducible conformal modules over \begin{document}$ \mathcal{W}(a,b,r) $\end{document}.

We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra \begin{document}$ V_{\mathcal{L}}(\ell_{123},0) $\end{document}. Then, for any integer \begin{document}$ t>1 $\end{document}, we introduce a new Lie algebra \begin{document}$ \mathcal{L}_{t} $\end{document}, and show that \begin{document}$ \sigma_{t} $\end{document}-twisted \begin{document}$ V_{\mathcal{L}}(\ell_{123},0) $\end{document}(\begin{document}$ \ell_{2} = 0 $\end{document})-modules are in one-to-one correspondence with restricted \begin{document}$ \mathcal{L}_{t} $\end{document}-modules of level \begin{document}$ \ell_{13} $\end{document}, where \begin{document}$ \sigma_{t} $\end{document} is an order \begin{document}$ t $\end{document} automorphism of \begin{document}$ V_{\mathcal{L}}(\ell_{123},0) $\end{document}. At the end, we give a complete list of irreducible \begin{document}$ \sigma_{t} $\end{document}-twisted \begin{document}$ V_{\mathcal{L}}(\ell_{123},0) $\end{document}(\begin{document}$ \ell_{2} = 0 $\end{document})-modules.