Simple Witt Modules that are Finitely Generated over the Cartan Subalgebra

Abstract

Let d ≥ 1 be an integer, W d and K d be the Witt algebra and the weyl algebra over the Laurent polynomial algebra, respectively. For any gl d -module M and any admissible module P over the extended Witt algebra W d , we define a W d -module structure on the tensor product P ⊗ M . We prove in this paper that any simple W d -module that is finitely generated over the cartan subalgebra is a quotient module of the W d -module P ⊗ M for a finite dimensional simple gl d -module M and a simple K d -module P that are finitely generated over the cartan subalgebra. We also characterize all simple K d -modules and all simple admissible W d -modules that are finitely generated over the cartan subalgebra.

2-Local derivations on Witt algebras

“…The Witt algebra W n of vector fields on an n-dimensional torus is the derivation Lie algebra of the Laurent polynomial algebra A n = C[t ±1 1 , t ±1 2 , · · · , t ±1 n ]. Over the last two decades, the representation theory of Witt algebras was extensively studied by many mathematicians and physicists; see for example [2,3,6]. Very recently, Billig and Futorny [2] obtained the classification for all simple Harish-Chandra W n -modules.…”

“…Over the last two decades, the representation theory of Witt algebras was extensively studied by many mathematicians and physicists; see for example [2,3,6]. Very recently, Billig and Futorny [2] obtained the classification for all simple Harish-Chandra W n -modules.…”

Local and $2$-local automorphisms of simple generalized Witt algebras

“…Using different notations from Kawamoto's, Djokovic and Zhao [10] gave an essentially equivalent definition of generalized Witt algebras. Over the last two decades, the representation theory of generalized Witt algebras over complex numbers was extensively studied by many mathematicians and physicists; see for example [3], [4] and [11]. Very recently, Billig and Futorny [3] obtained the classification for all simple Harish-Chandra modules over Lie algebra W n of vector fields on an n-dimensional torus, a well-known generalized Witt algebra.…”

Whittaker category for the Lie algebra of polynomial vector fields

“…associative. This kind of modules play an important role in the classification of simple modules for Cartan type Lie algebras, see [3,4,16,14,36]. For an a ∈ C n and a Whittaker W…”