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 $a\in \mathbb{C},{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n\in {\mathbb{C}}^{\ast }$, 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 $m\in {\mathbb{Z}}_{+}$. 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.

19 pages

References in corpus (2)

• Simple sl_{n+1}-module structures on U(h)
• Classification of simple $W_n$-modules with finite-dimensional weight spaces

Cited by in corpus (5)

• $\mathcal{U}(\mathfrak{h})$-free modules and coherent families
• Modules over the Heisenberg-Virasoro and $W(2,2)$ algebras
• Simple Witt modules that are finitely generated over the cartan subalgebra
• $\W_n^+$- and $W_n$-module structures on $U(h)$
• Simple modules over the Lie algebras of divergence zero vector fields on a torus