# THE NON-DEGENERACY OF THE SKEW-SYMMETRIC BILINEAR FORM OF THE FINITE DIMENSIONAL REAL FROBENIUS LIE ALGEBRA

### Abstract

*A** Frobenius Lie algebra is recognized as the Lie algebra whose stabilizer at a Frobenius functional is trivial. This condition is equivalent to the existence of a skew-symmetric bilinear form which is non-degenerate. On the other hand, the Lie algebra is Frobenius as well if its orbit on the dual vector space is open. In this paper, we study the skew-symmetric bilinear form of finite dimensional Frobenius Lie algebra corresponding to its Frobenius functional. The work aims to prove that a Lie algebra of dimension * * is Frobenius if and only if the * *-th derivation of the Frobenius functional is not equal to zero. Indeed, this condition implies that the skew-symmetric bilinear form is non-degenerate and vice versa. In addition, some properties of Frobenius functionals are obtained. Furthermore, the computations are given using the coadjoint orbits and the structure matrix. As a discussion, we can investigate these results **in** the * *algebra case whether giv**ing** rise to a left-invariant K* *hler structure of a Frobenius Lie group or not. *

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