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 $...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Naim Çağman
2023-06-01
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| Series: | Journal of New Theory |
| Subjects: | |
| Online Access: | https://dergipark.org.tr/en/download/article-file/3091247 |
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| Summary: | 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. |
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| ISSN: | 2149-1402 |