Lie algebra, nil

2020 Mathematics Subject Classification: Primary: 17B30 [MSN][ZBL]

A Lie algebra ${\mathfrak g}$ over a field $k$ is called nil if there is a function $n:{\mathfrak g}\times{\mathfrak g}\to{\mathbb N}$ such that $({\rm ad}\; x)^{n(x,y)}(y) = 0$, where $({\rm ad}\; x)(y) = [x,y]$, for any $x,y\in{\mathfrak g}$. The main question about nil Lie algebras concerns the conditions on ${\mathfrak g}$, $k$, $n$ under which ${\mathfrak g}$ is (locally) nilpotent (see Lie algebra, nilpotent). A nil Lie algebra that is finite-dimensional over $k$ is nilpotent. On the other hand, over any field there are finitely-generated nil Lie algebras that are not nilpotent [Go]. Suppose that $n$ is a constant. A nil Lie algebra is locally nilpotent if ${\rm char}\; k = 0$ or if $m\le p+1$, where $p={\rm char}\; k>0$ (Kostrikin's theorem, [Ko]). Local nilpotency also holds in the case when ${\mathfrak g}$ is locally solvable. An infinitely-generated nil Lie algebra is not necessarily nilpotent if $n\ge p-2$ (see [Ra]), and for $n\ge p+1$ non-nilpotency can still occur under the condition of solvability. Recently it has been proved by E.I. Zel'myanov that a nil Lie algebra is nilpotent if ${\rm char}\; k = 0$ (cf. [Ko2]) and that a nil algebra is also locally nilpotent if $n> p+1$. The study of nil Lie algebras over a field $k$ of characteristic $p>0$ is closely connected with the Burnside problem.

References

[Br]	A. Braun, "Lie rings and the Engel condition" J. of Algebra, 31 (1974) pp. 287–292 MR0344299 Zbl 0358.20051
[Go]	E.S. Golod, "On nil-algebras and residually finite groups" Izv. Akad. Nauk SSSR Ser. Mat., 28 : 2 (1964) pp. 273–276 (In Russian)
[Hu]	J.E. Humphreys, "Introduction to Lie algebras and representation theory", Springer (1972) pp. §5.4 MR0323842 Zbl 0254.17004
[Ja]	N. Jacobson, "Lie algebras", Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[Ko]	A.I. Kostrikin, "On Burnside's problem" Izv. Akad. Nauk SSSR Ser. Mat., 23 : 1 (1959) pp. 3–34 (In Russian)
[Ko2]	A.I. Kostrikin, "Around Burnside", Springer (1989) (Translated from Russian)
[Ra]	Yu.P. Razmyslov, "On Lie algebras satisfying the Engel condition" Algebra and Logic, 10 : 1 (1971) pp. 21–29 Algebra i Logika, 10 : 1 (1971) pp. 33–44 Zbl 0253.17005