Question

Let $\mathfrak{g}$ be a Lie Algebra (finite dimensional, over $\mathbb{C}$). Engel's theorem tells us that if there exists a $m\in \mathbb{N}$ such that $ad(x)^m = 0$, $\forall x\in \mathfrak{g}$, then $\mathfrak{g}$ is nilpotent. And if $\mathfrak{g}$ is $k$-step nilpotent (i.e. the $k$-th term of the lower central series of $\mathfrak{g}$ is the first one that is 0, or equivalently $ad(x_1)ad(x_2) \ldots ad(x_k) = 0$ $\forall x_1, \ldots, x_k \in \mathfrak{g}$), it is clear that $$\min(m\in \mathbb{N} : ad(x)^m = 0 \forall x\in \mathfrak{g} ) \leq k.$$ Can we find an example where the previous inequality is not an equality?

If this is a very basic fact in the theory, I apologize.

Answer 1

A Lie algebra $\mathfrak{g}$ satisfying the equation $ad(x)^m = 0$ for all $x \in \mathfrak{g}$ is called an Engel-$m$ Lie algebra. One of the key steps in Zelmanov's solution of the restricted Burnside problem is that any finitely generated Engel-$m$ Lie algebra is nilpotent. Zelmanov won the Fields medal for this work.

Let $E(t,m)$ be the free Lie algebra on $t$-generators subject to the Engel-$m$ identity. Then it follows that $E(t,m)$ is nilpotent and therefore finite dimensional, although its dimension and class depend on the characteristic of the ground field $k$.

Let us assume that $char(k) = 0$ for simplicity. Then it is not difficult to show that $E(t,2)$ is nilpotent of class $2$, and $E(2,3)$ is nilpotent of class $3$. The dimension of $E(t,2)$ is $t + \binom{t}{2}$, and the dimension of $E(2,3)$ is $5$.

But already $E(3,3)$ is actually nilpotent of class $4$, so this gives an example of a Lie algebra where the required inequality is strict.

More information can be found in the works of Michael Vaughan-Lee and Gunnar Traustason --- see, for example, http://people.bath.ac.uk/gt223/paper01.pdf. Traustason observes at the bottom of page 12 of this paper that "it is easy to construct an Engel-$3$ Lie algebra of class $4$".

I don't know the dimension of $E(3,3)$, but I'm sure that Willem de Graaf does. This paper contains more relevant information.

Answer 2

I computed the example of Traustason (see his remark). We start with the free-nilpotent Lie algebra of class 4 with 3 generators x1,x2,x3. It has dimension 32. Then we divide out the ideal generated by all brackets containg x1 three times, or x2,x3 containing at least 2-times. The quotient is an Engel-3-Lie algebra of nilpotency class 4 and of dimension 11. It is easy to write down explicit Lie brackets. This should be an example of least possible dimension. However, the next case, to find an Engel-4-Lie algebra of nilpotentcy class 7, of least possible dimension is more complicated, and I am frightened to do the calculation.