View source for Lie algebra, nil

{{MSC|17B30}}
{{TEX|done}}


A
[[Lie algebra|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|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
{{Cite|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,
{{Cite|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
{{Cite|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.
{{Cite|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|Burnside problem]].

====References====
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|valign="top"|{{Ref|Br}}||valign="top"| A. Braun, "Lie rings and the Engel condition" ''J. of Algebra'', '''31''' (1974) pp. 287–292  {{MR|0344299}}  {{ZBL|0358.20051}} 
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|valign="top"|{{Ref|Go}}||valign="top"| E.S. Golod, "On nil-algebras and residually finite groups" ''Izv. Akad. Nauk SSSR Ser. Mat.'', '''28''' : 2 (1964) pp. 273–276 (In Russian)   
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|valign="top"|{{Ref|Hu}}||valign="top"| J.E. Humphreys, "Introduction to Lie algebras and representation theory", Springer (1972) pp. §5.4  {{MR|0323842}}  {{ZBL|0254.17004}} 
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|valign="top"|{{Ref|Ja}}||valign="top"| N. Jacobson, "Lie algebras", Interscience (1962) ((also: Dover, reprint, 1979))  {{MR|0148716}} {{MR|0143793}}  {{ZBL|0121.27504}} {{ZBL|0109.26201}} 
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|valign="top"|{{Ref|Ko}}||valign="top"| A.I. Kostrikin, "On Burnside's problem" ''Izv. Akad. Nauk SSSR Ser. Mat.'', '''23''' : 1 (1959) pp. 3–34 (In Russian)   
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|valign="top"|{{Ref|Ko2}}||valign="top"| A.I. Kostrikin, "Around Burnside", Springer (1989) (Translated from Russian)   
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|valign="top"|{{Ref|Ra}}||valign="top"| 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}} 
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