Infinite Dimensional 3-Lie Algebras and Their Connections to Harish-Chandra Modules

TL;DR

This paper constructs infinite-dimensional 3-Lie algebras from commutative associative algebras, explores their derivation algebras related to Witt algebras, and reveals their representations include Harish-Chandra modules and modules of toroidal Lie algebras.

Contribution

It introduces new classes of infinite-dimensional 3-Lie algebras and connects their representations to Harish-Chandra modules and toroidal Lie algebras.

Findings

Constructed two types of infinite-dimensional 3-Lie algebras from commutative associative algebras.

Established that their inner derivation algebras relate to Witt algebras.

Found that regular representations of the second type are Harish-Chandra modules.

Abstract

In this paper we construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.