Wn+- and WnWn-module structures on U(hn)U(hn)

Let hnhn be the Cartan subalgebra of the Witt algebras Wn+=DerC[t1,t2,...,tn] and Wn=DerC[t1±1,t2±1,⋯,tn±1] where 1≤n≤∞1≤n≤∞. In this paper, we classify the modules over Wn+ and over WnWn which are free U(hn)U(hn)-modules of rank 1. These are the Wn+-modules Ω(Λn,a,S)Ω(Λn,a,S) for some Λn=(λ1,⋯,λn)∈(C⁎)nΛn=(λ1,⋯,λn)∈(C⁎)n, a∈Ca∈C, and S⊂{1,2,...,n}S⊂{1,2,...,n}; and WnWn-modules Ω(Λn,a)Ω(Λn,a) for some Λn∈(C⁎)nΛn∈(C⁎)n and some a∈Ca∈C.