Z x Z-graded Lie algebras generated by the Virasoro algebra and $s{l}_2$

Abstract

The present paper is another step toward the classification of $\mathbb{Z}\times \mathbb{Z}$-graded Lie algebras: we classify $\mathbb{Z}\times \mathbb{Z}$-graded Lie algebras $A={\oplus }_{i,j\in \mathbb{Z}}{A}_{i,j}$ over a field $F$ of characteristic 0 with dim $A_{i,j}\le 1$ for $i,j\in \mathbb{Z}$, satisfying: (1) ${\mathring{A}}_0={\oplus }_{i\in \mathbb{Z}}{\mathring{A}}_{i,0}$ is isomorphic to the centerless Virasoro algebra, (2) $A_{0,-1},A_{0,0},A_{0,1}$ span the 3-dimensional simple Lie algebra $s{l}_2$, (3) $A$ is generated by the centerless Virasoro algebra and $s{l}_2$. These algebras include some rank 2 centerless Virasoro algebras, and some rank 2 Block algebras and their extensions.

13 pages, latex