On the Orbit Problem of Free Lie Algebras

Year 2023, Issue: 43, 83 - 91, 30.06.2023

Zeynep Yaptı Özkurt

https://doi.org/10.53570/jnt.1284897

https://izlik.org/JA42UB34ML

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

References

• J. H. C. Whitehead, \emph{On Equivalent Sets of Elements in a Free Group}, Annals of Mathematics 37 (4) (1936) 782--800.
• S. Gersten, \emph{On Whitehead’s Algorithm}, Bulletin of the American Mathematical Society 10 (2) (1984) 281--284.
• P. V. Silva, P. Weil, \emph{Automorphic Orbits in Free Groups: Words Versus Subgroups}, International Journal of Algebra and Computation 20 (4) (2010) 561{--}590.
• P. Brinkmann, \emph{Detecting Automorphic Orbits in Free Groups}, Journal of Algebra 324 (5) (2010) 1083--1097.
• A. G. Myasnikov, V. Shpilrain, \emph{Automorphic Orbits in Free Groups}, Journal of Algebra 269 (1) (2003) 18--27.
• D. Kozen, \emph{Complexity of Finitely Presented Algebras}, in: J. E. Hopcroft, E. P. Friedman, M. A. Harrison (Eds.), STOC '77: Proceedings of the Ninth Annual ACM Symposium on Theory of Computing, Colorado, 1977, pp. 164--177.
• Y. Bahturin, A. Olshanskii, \emph{Filtrations and Distortion in Infinite-Dimensional Algebras}, Journal of Algebra 327 (1) (2011) 251--291
• R. H. Fox, \emph{Free Dif and only iferential Calculus. I: Derivation in the Free Group Ring}, Annals of Mathematics 57 (3) (1953) 547--560.
• V. Shpilrain, \emph{On the Rank of an Element of a Free Lie Algebras}, Proceedings of the American Mathematical Society 123 (5) (1995) 1303{--}1307.
• P. M. Cohn, Free Rings and Their Relations, 2nd Edition, Academic Press, London, 1985.
• A. A. Mikhalev, A. A. Zolotykh, \emph{Rank\; and\; Primitivity\; of\; Elements\; of\; Free\; Colour\; Lie\; (p-)Superalgebras}, International Journal of Algebra and Computation 4 (4) (1994) 617--656.
• P. M. Cohn, \emph{Subalgebras of Free Associative Algebras}, Proceedings of the London Mathematical Society 14 (3) (1964) 618--632.
• V. Drensky, \emph{Automorphisms of Relatively Free Algebras}, Communications in Algebra 18 (12) (1990) 4323--4351.
• A. V. Yagzhev, \emph{Endomorphisms of Free Algebras}, Siberian Mathematical Journal 21 (1) (1980) 133--141.
• G. P. Kukin, \emph{Primitive Elements of Free Lie Algebras}, Algebrai-Logika 9 (4) (1970) 458--472.