KILLING FORM ALJABAR LIE FROBENIUS BERDIMENSI ≤4 UNTUK MENENTUKAN KESEMISEDERHANAANNYA

Abstract

We study the notion of the Killing form for Frobenius Lie algebras of dimension . The Killing form is a symmetric bilinear form on a finite dimensional Lie algebra  over a field  defined by  where  is denoted the trace and  is an adjoint representation of .  A Lie algebras is said to be semisimple if it has the nondegenerate Killing form. The research aims to consider the criterion for semisimplicity of Frobenius Lie algebras of dimension  by using the Killing form. The results show that each Frobenius Lie algebra of dimension  and   is not semisimple since the the Killing form is degenerate or in other words, a determinant of a representation matrix  of the Killing form is equal to zero. For the future research, it is still an open problem to consider the general formulas of the Killing form for higher dimensional Frobenius Lie algebra whether degenerate or nondegenerate such that the semisimplicity of a Lie algebra can be considered. We conjecture that each finite dimensional real Frobenius Lie algebra is not semisimple.