Generalization of One of Lie’s Theorems

The theory of Lie algebras has many applications in mathematics and physics. One possible way of generalizing the theory of Lie algebras is to develop the theory of Lie-like algebras algebras, where the notion of a Lie-like algebras algebra was introduced in [4]. One of Lie’s Theorems claims that the only irreducible representations of a solvable Lie algebra over an algebraically closed field k of characteristic 0 have dimension 1. We call this theorem Lie’s Theorem for convenience in this paper. Since Lie’s Theorem is one of the fundamental results in the theory of Lie algebras, finding the counterpart of Lie’s Theorem in the context of Lie-like algebras is of importance to develop the theory of Lie-like algebras. The purpose of this paper is to prove a generalization of Lie’s Theorem in the context of Lie-like algebras.