Classification of Quasifinite Modules over the Lie Algebras of Weyl Type

By Yu-cai Su. 2003

47 citations recorded by the source corpus.

Abstract For a nondegenerate additive subgroup Γ 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 (Γ,n) spanned by all differential operators uD1m1⋯Dnmn for u∈ F [Γ] (the group algebra), and m1,…,mn⩾0, where D1,…,Dn 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 (Γ,n) -module is a module of the intermediate series and a complete classification of quasifinite W (Γ,n) -modules is also given, if Γ is not isomorphic to Z .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ef4f55a5cf32a918dee6c193c1fed380b53907f3

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Citations in the corpus

Quasi-Lie algebras (2004, background): “We would also like to note that in the works [5, 28, 43, 44], in the case of usual derivations on Laurent polynomials, it has been specifically noted that a Lie bracket can be defined by expressions somewhat resembling a special case of (3.”
Quasifinite representations of a Lie algebra of Block type (2004, background): “; among them the most fundamental one is the W1+∞ algebra, whose representation theory, of interest to both mathematicians and physicists, has been well-developed (see, for example, [2,3,6–10,15]).”
Deformations of Lie Algebras using σ-Derivations (2004, background): “We would also like to note that in the works [5,28,42,43], in the case of usual derivations on Laurent polynomials, it has been specifically noted that a Lie bracket can be defined by expressions somewhat resembling a special case of (20).”