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Classification of simple bounded weight modules of the Lie algebra of vector fields on \(\mathbb{C}^n\). (English) Zbl 1512.17042

Letting \(W_n^+\) be the Lie algebra of the Lie algebra of vector fields on \(\mathbb{C}^n\), Y. Xue and R. Lü classify all simple bounded weight \(W_n^+\)-modules. Any such module is isomorphic to the simple quotient of a tensor module \(P\otimes M\) for a simple weight module \(P\) over the Weyl algebra \(K_n^+=\mathbb{C}[t_1,\ldots,t_n,\frac{\partial}{\partial t_1},\ldots, \frac{\partial}{\partial t_n}]\) and a finite-dimensional simple \(\mathfrak{gl}_n(\mathbb{C})\) module \(M\). More precisely, any simple bounded weight \(W_n^+\) module \(V\) is isomorphic to one of the following simple bounded \(W_n^+\)-weight modules: the one-dimensional trivial module, the tensor module \(P\otimes M\), where \(M\) is a simple finite-dimensional \(\mathfrak{gl}_n\)-module that is not isomorphic to \(V(\delta_l,l), l=0,1,\dots,n\), or \(L_n(P,l)\), where \(l\in \{1,2,\dots,n\}\) and \(P\) is a simple weight \(K_n^+\) module.

MSC:

17B66 Lie algebras of vector fields and related (super) algebras
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