Classification of Quasifinite Modules over the Lie Algebras of Weyl Type

Details Classification of Quasifinite Modules over the Lie Algebras of Weyl Type Classification of Quasifinite Modules over the Lie Algebras of Weyl Type

2003-04-03

arxiv.org/abs/math/0304033v1

Adv. Math., 174 (2003), 57-68

Mathematics

Quantum Algebra

11 pages, LaTeX

Scientific paper

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 $W(\Z,1)$-module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly bounded $W(\Z,1)$-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 $\Z$.

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