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[[Lie algebra|Lie algebra]] ${\mathfrak g}$ over a field $k$ defined by the
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[[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
presence of 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 ${\mathfrak g}$,
 
main question about nil Lie algebras concerns the conditions on ${\mathfrak g}$,
 
$k$, $n$ under which ${\mathfrak g}$ is (locally) nilpotent (see
 
$k$, $n$ under which ${\mathfrak g}$ is (locally) nilpotent (see
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over any field there are finitely-generated nil Lie algebras that are
 
over any field there are finitely-generated nil Lie algebras that are
 
not nilpotent
 
not nilpotent
[[#References|[1]]]. Suppose that $n$ is a constant. A nil Lie algebra
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{{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,
 
is locally nilpotent if ${\rm char}\; k = 0$ or if $m\le p+1$, where $p={\rm char}\; k>0$ (Kostrikin's theorem,
[[#References|[2]]]). Local nilpotency also holds in the case when ${\mathfrak g}$
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{{Cite|Ko}}). Local nilpotency also holds in the case when ${\mathfrak g}$
 
is locally solvable. An infinitely-generated nil Lie algebra is not
 
is locally solvable. An infinitely-generated nil Lie algebra is not
 
necessarily nilpotent if $n\ge p-2$ (see
 
necessarily nilpotent if $n\ge p-2$ (see
[[#References|[3]]]), and for $n\ge p+1$ non-nilpotency can still occur under
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{{Cite|Ra}}), and for $n\ge p+1$ non-nilpotency can still occur under
 
the condition of solvability. Recently it has been proved by
 
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.
 
E.I. Zel'myanov that a nil Lie algebra is nilpotent if ${\rm char}\; k = 0$ (cf.
[[#References|[6]]]) and that a nil algebra is also locally nilpotent
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{{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
 
if $n> p+1$. The study of nil Lie algebras over a field $k$ of
 
characteristic $p>0$ is closely connected with the
 
characteristic $p>0$ is closely connected with the
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD
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{|
valign="top"> E.S. Golod, "On nil-algebras and residually finite
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|-
groups" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''28''' : 2 (1964)
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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}}
pp. 273–276 (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD
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|-
valign="top"> A.I. Kostrikin, "On Burnside's problem"
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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)  
''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''23''' : 1 (1959) pp. 3–34 (In
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|-
Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">
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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}}
Yu.P. Razmyslov, "On Lie algebras satisfying the Engel condition"
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|-
''Algebra and Logic'' , '''10''' : 1 (1971) pp. 21–29 ''Algebra i
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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}}
Logika'' , '''10''' : 1 (1971) pp. 33–44</TD></TR><TR><TD
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|-
valign="top">[4]</TD> <TD valign="top"> Yu. [Yu.A. Bakhturin]
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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) 
Bahturin, "Lectures on Lie algebras" , Akademie Verlag
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|-
(1978)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">
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|valign="top"|{{Ref|Ko2}}||valign="top"| A.I. Kostrikin, "Around Burnside", Springer (1989) (Translated from Russian)  
A. Braun, "Lie rings and the Engel condition" ''J. of Algebra'' ,
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|-
'''31''' (1974) pp. 287–292</TD></TR><TR><TD valign="top">[6]</TD> <TD
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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}}
valign="top"> A.I. Kostrikin, "Around Burnside" , Springer (1989)
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|-
(Translated from Russian)</TD></TR></table>
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|}
 
 
 
 
 
 
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD
 
valign="top"> N. Jacobson, "Lie algebras" , Interscience (1962)
 
((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[a2]</TD>
 
<TD valign="top"> J.E. Humphreys, "Introduction to Lie algebras and
 
representation theory" , Springer (1972) pp. §5.4</TD></TR></table>
 

Latest revision as of 22:04, 5 March 2012

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
How to Cite This Entry:
Lie algebra, nil. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_nil&oldid=19574
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article