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 $(V, \phi_{\mu})$ of the Lie algebra ad$(A_{\omega}^{\delta})$, and prove that only $(V, \psi_{\lambda0})$ and $(V, \psi_{\lambda1})$ are induced modules.