On the Orbit Problem of Free Lie Algebras

By operationalizing $F_{n}$ as a free Lie Algebra of finite rank $n$, this work considers the orbit problem for $F_{n}$. The orbit problem is the following: given an element $u\in F_{n}$ and a finitely generated subalgebra $H$ of $F_{n}$, does $H$ meet the orbit of $u$ under the automorphism group $Aut F_{n}$ of $F_{n}$? It is proven that the orbit problem is decidable for finite rank $n$, $n\geqslant2$. Furthermore, we solve a particular instance of the problem -- i.e., whether $H$ contains a primitive element of $F_{n}$. In addition, some applications are provided. Finally, the paper inquires the need for further research.