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

## TL;DR

This paper explores the structure of modules over a specific 3-Lie algebra, establishing connections with induced modules of its derivation algebra and constructing infinite-dimensional modules with particular properties.

## Contribution

It introduces the concept of induced modules for 3-Lie algebras and constructs explicit infinite-dimensional modules, analyzing their relation to modules of the inner derivation algebra.

## Key findings

- Only certain modules are induced modules among the constructed infinite-dimensional modules.
- Established a relationship between 3-Lie algebra modules and modules of its derivation algebra.
- Constructed explicit examples of infinite-dimensional modules for the algebra.

## 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.