ℤ × ℤ‐Graded Lie Algebras Generated by the Virasoro Algebra and sl2

Abstract

The present paper is another step toward the classification of ℤ × ℤ-graded Lie algebras: we classify ℤ × ℤ-graded Lie algebras 𝒜 = ⊕i,j∈ℤ 𝒜i,j over a field 𝔽 of characteristic 0 with dim𝒜i,j ≤ 1 for i, j ∈ ℤ, satisfying: (1) 𝒜0 = ⊕i∈ℤ 𝒜i,0 ≃ Vir, the centerless Virasoro algebra, (2) 𝒜0,–1, 𝒜0,0, 𝒜0,1 span the 3-dimensional simple Lie algebra sl2, (3) 𝒜 is generated by the centerless Virasoro algebra and sl2. These algebras include some rank 2 centerless Virasoro algebras, and some rank 2 Block algebras and their extensions.