The construction of real Frobenius Lie algebras from non-commutative nilpotent Lie algebras of dimension

Abstract

In this present paper, we study real Frobenius Lie algebras constructed from non-commutative nilpotent Lie algebras of dimension ≤ 4. The main purpose is to obtain Frobenius Lie algebras of dimension ≤ 6. Particularly, for a given non-commutative nilpotent Lie algebras N of dimension ≤ 4 we show that there exist commutative subalgebras of dimension ≤ 2 such that the semi-direct sums ɡ = N⊕T is Frobenius Lie algebras. Moreover, T is called a split torus which is a commutative subalgebra of derivation of N and it depends on the given N. To obtain this split torus, we apply Ayala’s formulas of a Lie algebra derivation by taking a diagonal matrix of a standard representation matrix of the Lie algebra derivation of N. The discussion of higher dimension of Frobenius Lie algebras obtained from non-commutative nilpotent Lie algebras is still an open problem.