A characterization of certain loop Lie algebras
The authors classify \({\mathbb Z} \times {\mathbb Z}\)-graded Lie algebras \(A = \bigoplus_{i,j\in {\mathbb Z}} A_{ij}\) over a field of characteristic zero with \(\dim A_{i,j} \leqq 1\) such that \(A_{\pm 1,0}\) and \(A_{0,\pm 1}\) are one-dimensional and generate \(A_1 \bigoplus_{i\in {\mathbb Z}} A_{0,j} \cong s\ell_2\), \([A_{-1,0} ,A_{1,0} ] = 0\), and \(\operatorname {ad} A_{\pm 1,0}\) act faithfully on \(\bigoplus_{i\in {\mathbb Z}} A_{j,1}\). They show that \(A\) must be isomorphic to one of the two smaller loop algebras.
scientific article; zbMATH DE number 1384450