On the Orbit Problem of Free Lie Algebras

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Journal of New Theory
Issue:43
On the Orbit Problem of Free Lie Algebras

On the Orbit Problem of Free Lie Algebras

Pages : 83-91

View : 79 | Download : 42

Publication Date : 2023-06-30

Article Type : Research Paper

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

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