# Lie algebra, nil

A Lie algebra over a field defined by the presence of a function such that , where , for any . The main question about nil Lie algebras concerns the conditions on , , under which is (locally) nilpotent (see Lie algebra, nilpotent). A nil Lie algebra that is finite-dimensional over is nilpotent. On the other hand, over any field there are finitely-generated nil Lie algebras that are not nilpotent [1]. Suppose that is a constant. A nil Lie algebra is locally nilpotent if or if , where (Kostrikin's theorem, [2]). Local nilpotency also holds in the case when is locally solvable. An infinitely-generated nil Lie algebra is not necessarily nilpotent if (see [3]), and for 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 (cf. [6]) and that a nil algebra is also locally nilpotent if . The study of nil Lie algebras over a field of characteristic is closely connected with the Burnside problem.

#### References

[1] | E.S. Golod, "On nil-algebras and residually finite groups" Izv. Akad. Nauk SSSR Ser. Mat. , 28 : 2 (1964) pp. 273–276 (In Russian) |

[2] | A.I. Kostrikin, "On Burnside's problem" Izv. Akad. Nauk SSSR Ser. Mat. , 23 : 1 (1959) pp. 3–34 (In Russian) |

[3] | 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 |

[4] | Yu. [Yu.A. Bakhturin] Bahturin, "Lectures on Lie algebras" , Akademie Verlag (1978) |

[5] | A. Braun, "Lie rings and the Engel condition" J. of Algebra , 31 (1974) pp. 287–292 |

[6] | A.I. Kostrikin, "Around Burnside" , Springer (1989) (Translated from Russian) |

#### Comments

#### References

[a1] | N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) |

[a2] | J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4 |

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Lie algebra, nil.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_nil&oldid=13081