On the Orbit Problem of Free Lie Algebras
Authors : Zeynep Yapti Özkurt
Pages : 83-91
Doi:10.53570/jnt.1284897
View : 48 | Download : 42
Publication Date : 2023-06-30
Article Type : Research Article
Abstract :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.Keywords : Orbit, automorphism, free Lie algebras