Representations of n-Lie algebras

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 $tπ_1$ for some nonnegative integer t.