ON THE ALGEBRA OF CONSTANTS OF FREE METABELIAN LIE ALGEBRAS

Year 2022, Volume: 5 Issue: 2, 185 - 192, 31.07.2022

Andre Dushımırımana

https://doi.org/10.33773/jum.1143787

Abstract

Let $K$ be a field of characteristic zero, $X_n=\{x_1,\dots,x_n\}$ be a set of variables, $K[X_n]$ be the polynomial algebra and $F_n$ be the free metabelian Lie algebra of rank $n$ generated by $X_n$ over the base field $K$. Well known result of Weitzenb\"ock states that $K[X_n]^\delta=\big \{u\in K[X_n] \big\vert\ \delta(u)=0\big \}$ is finitely generated as an algebra, where $\delta$ is a locally nilpotent linear derivation of $K[X_n]$. Extending this ideal to the non commutative algebras, recently the algebra $F_n^\delta$ of constants in the free metabelian Lie algebras have been investigated. According to the findings, $F_n^\delta$ is not finitely generated as a Lie algebra; whereas, $F_n^\delta \cap F_n^\prime$ is finitely generated $K[X_n]^\delta$-module and a list of generators for $n\le 4$ was given. In this work, in filling the gap in the list of small $n'$s we work in $F_5$ and give a list of generators of $F_5^\delta$ where $\delta(x_5)=x_4$, $\delta(x_4)=0$, $\delta(x_3)=x_2$, $\delta(x_2)=x_1$ and $\delta(x_1)=0$.

Keywords

Lie Algebras, Algebras of Constants, Weitzenb¨ock Derivations

References

• Reference1 M. Nagata, On the 14-th problem of Hilbert, Amer. J. Math., 81 , 766-772 (1959).
• Reference2 E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann., 77, 89-92 (1916).
• Reference3 R. Weitzenbock, Über die Invarianten von linearen Gruppen, Acta Math., 58, 231-293 (1932).
• Reference4 W. Dicks, E. Formanek, Poincare Series and a problem of S. Montgomery, Linear Multilinear Algebra, 12, 21-30 (1982).
• Reference5 V.K. Kharchenko, Algebra of Invariants of Free Algebras (Russian), Algebra iLogika, 17, 478-487, Translation: Algebra and Logic,(1978) 17, 316-321 (1978).
• Reference6 R.M. Bryant, On the fixed points of a finite group acting on a free Lie algebra, J. London Math. Soc. 43 (2) 215-224 (1991).
• Reference7 V. Drensky, Fixed algebras of residually nilpotent Lie algebras, Proc. Amer. Math. Soc. 120 (4) 1021-1028 (1994).
• Reference8 R. Dangovski, V. Drensky, Ş. Fındık, Weitzenböck derivations of free metabelian Lie algebras, Linear Algebra and its Applications, 439 10, 3279-3296 (2013).
• Reference9 Yu.A. Bahturin, Identical Relations in Lie Algebras (Russian), ”Nauka”, Moscow, 1985. Translation: VNU Science Press, Utrecht, 1987.
• Reference10 A. Nowicki, Polynomial Derivations and Their Rings of Constants, Uniwersytet Mikolaja Kopernika, Torun, 1994. www-users.mat.umk.pl/\~{}anow/ps-dvi/pol-der.pdf.