Classification of Quasifinite Modules over the Lie Algebras of Weyl Type

Abstract

For a nondegenerate additive subgroup $G$ of the $n$-dimensional vector space $F^n$ over an algebraically closed field $F$ of characteristic zero, there is an associative algebra and a Lie algebra of Weyl type $W(G,n)$ spanned by all differential operators $u{D}_1^{m_1}...D_n^{m_n}$ for $u\in F[G]$ (the group algebra), and $m_1,...,m_n\ge 0$, where $D_1,...,D_n$ are degree operators. In this paper, it is proved that an irreducible quasifinite $W(\mathbb{Z},1)$-module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly bounded $W(\mathbb{Z},1)$-modules is completely given. It is also proved that an irreducible quasifinite $W(G,n)$-module is a module of the intermediate series and a complete classification of quasifinite $W(G,n)$-modules is also given, if $G$ is not isomorphic to $\mathbb{Z}$.

11 pages, LaTeX

References in corpus (3)

• Quasifinite highest weight modules over the Lie algebra of differential operators on the circle
• W_{1+\infty} and W(gl_N) with central charge N
• Determinant and Character of W-infinity algebra