Formula for dimension of graded component of free Lie algebra

Statement

Suppose $n$ and $r$ are positive integers. Consider the free Lie algebra over $\mathbb {Q}$ with $n$ generators. Further, consider the degree $r$ graded component of this free Lie algebra. This is a $\mathbb {Q}$ -vector space and is spanned by all the Lie brackets of length $r$ involving the generators. The dimension is equal to the number of equivalence classes of words of length $r$ in $n$ letters under cyclic permutations that have the property that no two cyclic permutations of the word are equal.

The expression is:

${\frac {1}{r}}\sum _{d\mid r}\mu (d)n^{r/d}$

where $\mu$ denotes the Mobius function.

Note that instead of using free $\mathbb {Q}$ -Lie algebras, we could also have used the free $\mathbb {Z}$ -Lie ring. In this case, it would turn out that each graded component is a free abelian group and the above formula gives the rank of that group.

Related facts

Nilpotent multiplier of abelian group is graded component of free Lie ring: Combined with this fact, we can compute the nilpotent multipliers of free abelian groups.
Basic Lie products form a freely generating set of graded component of free Lie algebra: This provides an explicit freely generating set (or basis, if we choose to work over a field)
Induction formula for Lie operad

Particular cases

Each row corresponds to a value of $r$ . For a given value of $r$ , the dimension is a polynomial in $n$ of degree $r$ . The polynomial is listed, then its values for small $n$ are given.

There are two ways that more detail could be provided. One is to provide the explicit Lyndon basis -- for more on that, see the basic Lie product page. The other is to provide explicit decompositions in terms of Schur functors. For more on that, see induction formula for Lie operad.

Value of $r$	Polynomial sending $n$ to dimension	Dimension case $n=1$	Dimension case $n=2$	Dimension case $n=3$	Dimension case $n=4$	Dimension case $n=5$
1	$n$	1	2	3	4	5
2	${\frac {n^{2}-n}{2}}$	0	1	3	6	10
3	${\frac {n^{3}-n}{3}}$	0	2	8	20	40
4	${\frac {n^{4}-n^{2}}{4}}$	0	3	18	60	150
5	${\frac {n^{5}-n}{5}}$	0	6	48	204	624
6	${\frac {n^{6}-n^{3}-n^{2}+n}{6}}$	0	9	116	670	2580
7	${\frac {n^{7}-n}{7}}$	0	18	312	2340	11160
8	${\frac {n^{8}-n^{4}}{8}}$	0	30	810	8160	48750
9	${\frac {n^{9}-n^{3}}{9}}$	0	56	2184	29120	217000
prime $p$	${\frac {n^{p}-n}{p}}$	0	${\frac {2^{p}-2}{p}}$	${\frac {3^{p}-3}{p}}$	${\frac {4^{p}-4}{p}}$	${\frac {5^{p}-5}{p}}$
prime power $p^{t}$	${\frac {n^{p^{t}}-n^{p^{t-1}}}{p^{t}}}$	0	${\frac {2^{p^{t}}-2^{p^{t-1}}}{p^{t}}}$	${\frac {3^{p^{t}}-3^{p^{t-1}}}{p^{t}}}$	${\frac {4^{p^{t}}-4^{p^{t-1}}}{p^{t}}}$	${\frac {5^{p^{t}}-5^{p^{t-1}}}{p^{t}}}$
product $p_{1}p_{2}$ of distinct primes	${\frac {n^{p_{1}p_{2}}-n^{p_{1}}-n^{p_{2}}+n}{p_{1}p_{2}}}$	0	${\frac {2^{p_{1}p_{2}}-2^{p_{1}}-2^{p_{2}}+2}{p_{1}p_{2}}}$	${\frac {3^{p_{1}p_{2}}-3^{p_{1}}-3^{p_{2}}+3}{p_{1}p_{2}}}$	${\frac {4^{p_{1}p_{2}}-4^{p_{1}}-4^{p_{2}}+4}{p_{1}p_{2}}}$	${\frac {5^{p_{1}p_{2}}-5^{p_{1}}-5^{p_{2}}+5}{p_{1}p_{2}}}$