Irreducible modules over Witt algebras $\mathcal{W}_n$ and over $\mathfrak{sl}_{n+1}(\mathbb{C})$

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 the necessary and sufficient conditions for $A_b$ to be irreducible, and determine the necessary and sufficient conditions for two such irreducible $\mathcal{W}_n$-modules to be isomorphic. Since $\sl_{n+1}(\mathbb{C})$ is a subalgebra of $\mathcal{W}_n$, all the above irreducible $\mathcal{W}_n$-modules $A_b$ can be considered as $\sl_{n+1}(\mathbb{C})$-modules. For a class of such $\sl_{n+1}(\mathbb{C})$-modules, denoted by $\Omega_{1-a}(\lambda_1,\lambda_2,\cdots,\lambda_n)$ where $a\in\mathbb{C}, \lambda_1,\lambda_2,\cdots,\lambda_n \in \mathbb{C}^*$, we determine the necessary and sufficient conditions for these $\sl_{n+1}(\mathbb{C})$-modules to be irreducible. If the $\sl_{n+1}(\mathbb{C})$-module $\Omega_{1-a}(\lambda_1,\lambda_2,\cdots,\lambda_n)$ is reducible, we prove that it has a unique nontrivial submodule $W_{1-a}(\lambda_1, \lambda_2,...\lambda_n)$ and the quotient module is the finite dimensional $\sl_{n+1}(\mathbb{C})$-module with highest weight $m\Lambda_n$ for some non-negative integer $m\in \Z_+$. The necessary and sufficient conditions for two $\mathfrak{sl}_{n+1}(\mathbb{C})$-modules $\Omega_{1-a}(\lambda_1,\lambda_2,\cdots,\lambda_n)$ and $W_{1-a}(\lambda_1, \lambda_2,...\lambda_n)$ to be isomorphic are also determined. The irreducible $\mathfrak{sl}_{n+1}(\mathbb{C})$-modules $\Omega_{1-a}(\lambda_1, \lambda_2,...\lambda_n)$ and $W_{1-a}(\lambda_1, \lambda_2,...\lambda_n)$ are new.