Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field of characteristic zero, a Lie algebra (G) of Block type is defined with basis {L _α,i | α ∈ G, i ∈ ℤ₊} and relations L _α,i , L _β,j ] = (β − α)L _α+β,i+j (αj − βi)L _{α+β,i+j−1}. It is proved that an irreducible highest weight (ℤ)-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order ≻ on G and any Λ ∈ (G)₀^* (the dual space of (G)₀ = span{L _0,i | i ∈ ℤ₊}), a Verma (G)-module M(Λ, ≻) is defined, and the irreducibility of M(Λ, ≻) is completely determined. Supported by NSF Grant No. 10471091 of China, the Grant of “One Hundred Talents Program” from the University of Science and Technology of China