Irreducible cuspidal modules of simple $n$-Lie algebras

TL;DR

This paper classifies irreducible cuspidal modules over certain simple n-Lie algebras, focusing on Wronskian and Jacobian types, and demonstrates their irreducibility across these structures.

Contribution

It provides a detailed classification of irreducible cuspidal modules over Wronskian n-Lie algebras and proves their irreducibility over Jacobian n-Lie algebras, expanding understanding of module structures.

Findings

Classified irreducible cuspidal modules over Wronskian n-Lie algebras.

Proved modules remain irreducible over Jacobian n-Lie algebras.

Analyzed Lie and Leibniz structures on wedge and tensor powers of n-Lie algebras.

Abstract

This work devoted to the description of irreducible cuspidal modules over simple $n$-Lie algebras. Since the description of irreducible modules over $n$-Lie algebra $O^n$ are already well understood, we focus here on the irreducible cuspidal modules over $n$-Lie algebras of Wronskians and Jacobians. First, for a given $n$-Lie algebra $\mathcal{L}$, we analyze the possible Lie and Leibniz structures on $\wedge^{n-1} \mathcal{L}$ and $\otimes^{n-1} \mathcal{L}$ by thoroughly examining existing structures. Next, we classify the irreducible cuspidal modules over the $n$-Lie algebra of Wronskians defined on Laurent polynomials with degree-preserving derivations. Furthermore, we prove that these modules remain irreducible over the $n$-Lie algebra of Jacobians.