Representations of n-Lie algebras

TL;DR

This paper studies the structure of finite-dimensional modules over n-Lie algebras, showing they are completely reducible and classifying irreducible modules via extensions of so_{n+1}-modules with specific highest weights.

Contribution

It provides a classification of finite-dimensional irreducible modules over n-Lie algebras as extensions of so_{n+1}-modules with explicit highest weights.

Findings

Finite-dimensional n-Lie modules are completely reducible.

Irreducible n-Lie modules are isomorphic to extensions of so_{n+1}-modules.

Highest weight of irreducible modules is tπ_1 for some nonnegative integer t.

Abstract

Let $V_n=<e_1,...,e_{n+1}>$ be a vector products n-Lie algebra with n-Lie commutator $[e_1,...,\hat{e_i},...,e_{n+1}]=(-1)^ie_i$ over the field of complex numbers. Any finite-dimensional n-Lie $V_n$-module is completely reducible. Any finite-dimensional irreducible n-Lie $V_n$-module is isomorphic to a n-Lie extension of $so_{n+1}$-module with highest weight $t\pi_1$ for some nonnegative integer t.