Classification of irreducible representations of Lie algebra of vector fields on a torus

Abstract

We solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on

Lie algebras of vector fields on smooth affine varieties

“…It follows from part (2) that a relation [ν, η] = λη with λ ∈ k, λ = 0, can not hold for non-zero elements in D H . This proves (3).…”

New irreducible weight modules over Witt algebras with infinite-dimensional weight spaces

“…We have to mention the remarkable work by Billig and Futorny [3] in which they proved very recently that irreducible modules for W d with finite-dimensional weight spaces fall in two classes: (1) modules of the highest weight type and (2) irreducible modules from F α b (V ). So the next task in representation theory of Witt algebras is to study irreducible modules over the Witt algebra W d with infinite-dimensional weight spaces, and irreducible nonweight modules.…”

Classification of Simple Cuspidal Modules Over a Lattice Lie Algebra of Witt Type

“…By Corollary 3.4 of [BF1], it is known that, for any cuspidal Wmodule V whose multiplicity is bounded by a constant, say d, there is m ∈ Z >0 (determined by d) such that the differentiator Ω (m) rξ,sξ;ξ acts trivially on V , for any r, s ∈ Z. With the aid of this fact, we show the next proposition: Proposition 2.3.…”