3-Lie algebra $A_{\omega }^{\delta }$-modules and induced modules

Abstract

In this paper, we define the induced modules of Lie algebra ad$(B)$ associated with a 3-Lie algebra $B$-module, and study the relation between 3-Lie algebra $A_{\omega }^{\delta }$-modules and induced modules of inner derivation algebra ad$(A_{\omega }^{\delta })$. We construct two infinite dimensional intermediate series modules of 3-Lie algebra $A_{\omega }^{\delta }$, and two infinite dimensional modules $(V,{\psi }_{\lambda \mu })$ and $(V,{\varphi }_{\mu })$ of the Lie algebra ad$(A_{\omega }^{\delta })$, and prove that only $(V,{\psi }_{\lambda 0})$ and $(V,{\psi }_{\lambda 1})$ are induced modules.