The Charactherization Criterion of g-quasi-Frobenius Lie Algebra Corresponding to Inner Derivation

The structure of a g-quasi-Frobenius Lie Algebra can be realized as a quasi-Frobenius Lie Algebra modules over a Lie Algebra g. This research discusses a special condition of the g-quasi-Frobenius Lie Algebra, namely when g acts on its self. This condition supports the construction of an inner derivation on g. The criterion investigated is the characterization of the g-quasi-Frobenius Lie Algebra in relation to the inner derivation. The result obtained is a criterion: a g-quasi-Frobenius Lie Algebra can be constructed on g itself if and only if the inner derivation is zero. Furthermore, several concrete examples are provided to test this criterion.