A category equivalence on the Lie algebra of polynomial vector fields

Abstract

For any positive integer n, let Wn=Der(C[t1,…,tn]). The subspaces hn=Span{t1∂∂t1,…,tn∂∂tn} and Δn=Span{∂∂t1,…,∂∂tn} are two abelian subalgebras of Wn. We show that a full subcategory Ω1 of the category of Wn-modules M which are locally finite over Δn is equivalent to some full subcategory of weight Wn-modules M which are cuspidal modules when restricted to the subalgebra sln+1 of Wn.