Doubly \(\mathbb{Z}\)-graded Lie algebras containing a Virasoro algebra (Q1306839)

The authors continue their study of \(\mathbb{Z} \times \mathbb{Z}\)-graded algebras \(A=\bigoplus_{i,j\in \mathbb{Z}} A_{i,j}\) over a field of characteristic 0 where each space \(A_{i,j}\) has dimension at most 1. In this paper the authors suppose that \(A\) satisfies the further properties that \(\bigoplus_{i\in \mathbb{Z}} A_{0,i}\) is isomorphic to the Virasoro algebra; that \([A_{-1,0},A_{1,0}]=0\), and \(\text{ad} A_{-1,0}\) and \(\text{ad} A_{1,0}\) act faithfully on \(\bigoplus_{i\in \mathbb{Z}}A_{i,1}\), and that \(\dim A_{\pm 1,0}=\dim A_{0,\pm 1} = 1\) and \(A\) is generated by \(A_{\pm 1,0}\) and \(A'_0\). Under these conditions the authors show that the algebra \(A\) is isomorphic either to a Block algebra, or to a certain extension of a Block algebra.