Indecomposable modules of the intermediate series over W(a,b) algebras

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For any complex parameters a,b, the W(a,b) algebra is the Lie algebra with basis Lᵢ,Wᵢ|i∈ Z, and relations [Lᵢ,Lⱼ]=(j-i)Lᵢ₊ⱼ, [Lᵢ,Wⱼ]=(a+j+bi)Wᵢ₊ⱼ,[Wᵢ,Wⱼ]=0. In this paper, indecomposable modules of the intermediate series over W(a,b) are classified. It is also proved that an irreducible Harish-Chandra W(a,b)-module is either a highest/lowest weight module or a uniformly bounded module. Furthermore, if a∉ Q, an irreducible weight W(a,b)-module is simply a Vir-module with trivial actions of Wₖ.