Doubly Z-Graded Lie Algebras Containing a Virasoro Algebra

In this paper we study Z × Z-graded Lie algebras A = i, j ZAi, j over a characteristic 0 field F with dim Ai, j ≤ 1 for each i and j, where Z denotes the integers. We shall assume that A satisfies the following three properties:• (I)′ = Vir, the centerless Virasoro algebra;• (II)[, ] = 0, and ad and ad act faithfully on the subspace ;• (III)dim = dim = 1, and is generated by and ′. The main result of the paper is the classification of these algebras. More precisely, A is isomorphic to a simple Block algebra L(a) which has basis {ei, j (i, j) Z × Z\{(−1/a, 0), (−2/a, 0)}} and subject to the bracket[formula]for some suitable constant a F\{0, ± 1, ± 2}, or is isomorphic to a one- or two-dimensional central extension of L(a) with a F\{0, ± 1, ± 2}, or A is isomorphic a one-dimensional derivation extension of L( ± 1) and L( ± 2), or a two-dimensional extension of L( ± 1) by a center element and a derivation.