Irreducible Modules Over Witt Algebras W n $\mathcal {W}_{n}$ and Over 𝖘 𝖑 n + 1 ( ℂ ) $\mathfrak {sl}_{n+1}(\mathbb {C})$

Abstract

In this paper, by using the “twisting technique” we obtain a class of new modules A b over the Witt algebras $\mathcal {W}_{n}$ from modules A over the Weyl algebras $\mathcal {K}_{n}$ (of Laurent polynomials) for any $b\in \mathbb {C}$ . We give necessary and sufficient conditions for A b to be irreducible, and determine necessary and sufficient conditions for two such irreducible $\mathcal {W}_{n}$ -modules to be isomorphic. Since $\mathfrak {sl}_{n+1}(\mathbb {C})$ is a subalgebra of $\mathcal {W}_{n}$ , all the above irreducible $\mathcal {W}_{n}$ -modules A b can be considered as $\mathfrak {sl}_{n+1}(\mathbb {C})$ -modules. For a class of such $\mathfrak {sl}_{n+1}(\mathbb {C})$ -modules, denoted by Ω1−a (λ 1, λ 2, ⋯ ,λ n ) where $a\in \mathbb {C}, \lambda _{1},\lambda _{2},\cdots ,\lambda _{n} \in \mathbb {C}^{*}$ , we determine necessary and sufficient conditions for these $\mathfrak {sl}_{n+1}(\mathbb {C})$ -modules to be irreducible. If the $\mathfrak {sl}_{n+1}(\mathbb {C})$ -module Ω1−a (λ 1, λ 2,⋯ ,λ n ) is reducible, we prove that it has a unique nontrivial submodule W 1−a (λ 1, λ 2,...λ n ) and the quotient module is the finite dimensional $\mathfrak {sl}_{n+1}(\mathbb {C})$ -module with highest weight mΛ n for some non-negative integer $m\in \mathbb {Z}_{+}$ . We also determine necessary and sufficient conditions for two $\mathfrak {sl}_{n+1}(\mathbb {C})$ -modules of the form Ω1−a (λ 1, λ 2,⋯ ,λ n ) or of the form W 1−a (λ 1, λ 2,...λ n ) to be isomorphic.