Representations of Vector Product n-Lie Algebras

Abstract

Let V _n  = ⟨e ₁,…, e _n+1⟩ be the vector product n-Lie algebra with n-Lie commutator [e ₁,…, ˆe _i ,…, e _n+1] = (− 1) ⁱ e _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 an n-Lie extension of s o _n+1-module with highest weight tπ₁ for some nonnegative integer t.

Key Words:

• Vector products algebra
• Lie algebras
• n-Lie algebras
• Nambu algebras
• Representations
• N-Commutators

1991 Mathematics Subject Classification:

• Primary 17C50
• 17B66
• 22E65

Acknowledgments

I am grateful to INTAS foundation for support. I am thankful to referee for paying of my attention to the preprint Sh. M. Kasymov, E. N. Kuz'min, “Classification of representations of split simple n-Lie algebra of type A ₁”, preprint No. 23, IM SO RAN, Novosibirsk, 1987, where was obtained results similar to ours.

Notes

^#Communicated by I. Shestakov.