# Lie algebra, nil

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.