# Free Lie algebra and nilpotent groups in Rothschild and Stein's paper

In

Rothschild, Linda Preiss; Stein, Elias M., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137(1976), 247-320 (1977). ZBL0346.35030. PDF at archive.ymsc.tsinghua.edu.cn

they mentioned the free nilpotent Lie algebra $$\mathfrak{R}_{F,s}$$ in Part I, section 3, example 4. And they introduced $$n_s$$ and $$m_s$$ in Part II, section 7, where

• $$n_s$$ is the dimension of the free nilpotent Lie algebra $$\mathfrak{R}_{F,s}$$ of step $$s$$ on $$n$$ generators,
• $$m_s$$ is the dimension of the linear space spanned by all commutators of the vector fields $$\{W_k\}_{k=1}^n$$ ($$n$$ is the number of vector fields) of lengths $$\leq s$$ restricted to $$\xi$$.

I can understand $$m_s$$ easily but It's difficult to understand $$n_s$$ to me. I think I need some example for $$n_s$$. Consider the vector fields: $$X=(X_1,X_2)=(\partial_1,x_1\partial_2)$$ Its $$m_1=1$$ restricted to point $$(0,x_2)$$ and $$m_1=2$$ at others, while $$m_2=2$$ at any point in $$\mathbb{R}^2$$. What is $$n_1$$ and $$n_2$$ for this example? why?

And by the theorem 4 and introduction in this paper, if we lift the vector fields as $$\widetilde{X}=(\widetilde{X_1},\widetilde{X_2})=(\partial_1,x_1\partial_2+\partial_3)$$ then, $$m_2=3$$ and $$m_2=n_2$$. Why does $$m_2=n_2$$?

• You don't introduce $W_k$. Also I don't see the meaning of "its $m_1=1$"... Also calling $n_s$ a function of $n$ and $s$ is utterly confusing.
– YCor
Jan 5 '20 at 23:27
• Sorry. $\{W_k\}_{k=1}^{m}$ are vector fields, so are the $X$ below. The definition of $n_s$ and $m_s$ is the original words in his paper. In the first example, when $s=1$, then at the point $(0,x_2)$ we can see $X=(\partial_1,0)$. Thus $m_1=dim(span\{\partial_1,0\})=1$. The other cases are similar. But I don't understand the definition of $n_s$ in his paper, even don't know how to calculate $n_s$ for this example.
– Houa
Jan 6 '20 at 1:55
• More explanations. When $s=2$, we have the commutator $[X_1,X_2]=\partial_2$, thus by the definition, we have $m_2=dim(span\{\partial_1,x_1\partial_2,\partial_2\})=2$ at any point in $\mathbb{R}^2$.
– Houa
Jan 6 '20 at 2:02
• @YCor: The notation is indeed a bit unfortunate. I think in context, R&S are treating $n$ as fixed throughout, and the $n$ in $n_s$ is not a reference to the same $n$. Jan 6 '20 at 8:22

First, the notation is a little confusing. I think you are supposed to understand that $$n$$ is always the number of vector fields in the fixed set $$\{X_1, \dots, X_n\}$$. But the symbol $$n$$ in the notation $$n_s$$ is just an arbitrary letter and isn't a reference to the number of vector fields. So in your example, $$n$$ is $$2$$, and $$n_1$$ just means "the dimension of the free nilpotent algebra of step 1 on 2 generators". $$n_2$$ is the dimension of the free nilpotent algebra of step 2 on 2 generators. If we were working with a set of 47 vector fields, then $$n_2$$ would refer to the dimension of the free nilpotent algebra of step 2 on 47 generators.

In small examples, this is easy to compute. $$n_1$$ will always equal $$n$$, because the free nilpotent Lie algebra of step $$1$$ on $$n$$ generators, call them $$Y_1, \dots, Y_n$$, is simply the abelian Lie algebra spanned by $$Y_1, \dots, Y_n$$ with all brackets vanishing, and its dimension is $$n$$. In your example with $$n=2$$, we have $$n_2 = 3$$; the free nilpotent Lie algebra of step $$2$$ on $$2$$ generators is the Heisenberg Lie algebra spanned by $$Y_1, Y_2, Z$$, where $$[Y_1, Y_2]=Z$$ and $$[Y_1, Z]=[Y_2, Z]=0$$.

When $$s \ge 3$$ it gets harder. You can't just count all possible brackets of $$Y_1, \dots, Y_n$$ of order up to $$s$$, because Jacobi's identity implies some linear dependence among them. But in general, the value of $$n_s$$ can be found from Witt's formula; see Corollary 4.14 of

Reutenauer, Christophe, Free Lie algebras, London Mathematical Society Monographs. New Series. 7. Oxford: Clarendon Press. xvii, 269 p. (1993). ZBL0798.17001.

Indeed, we have $$n_s = \sum_{k=1}^s \frac{1}{k} \sum_{d \mid k} \mu(d) n^{k/d}$$ where $$\mu$$ is the Möbius function.

• The Witt's formula seems that ns only depends on the number of the generators. By the description in R&S's paper, I try to explain like the following. For given "generator": $Y_1,Y_2,Y_3$, let them generate the "free nilpotent Lie algebra" $\mathfrak{R}_{F,2}$ (let, say $s=2$,) $=\{Y_1,Y_2,Y_3,[Y_1,Y_2],[Y_1,Y_3],[Y_2,Y_3]\}$. Thus $n_2=6$, which is consistent with the result made by Witt's formula. Once we set $Y_i$ to be a specific vector field, like $Y_1=\partial_1$ etc. then we can calculate $m_2$ at some point and obviously $m_2\leq n_2$.
– Houa
Jan 7 '20 at 8:08
• @xixixi: The value of $n_s$ given by Witt depends on both the number of generators $n$ and the step $s$ (the sum on $k$ is up to $s$). But yes, your explanation is basically right. $n_s$ is the "worst case" dimension, where none of the brackets coincide with each other, and so $m_s$ can be no larger. Jan 7 '20 at 17:01