Basic commutator

regular commutator

An object inductively constructed from the elements of a given set $R$ and from brackets, in the following manner. The elements of $R$ are considered by definition to be basic commutators of length 1, and they are given an arbitrary total order. The basic commutators of length $n$, where where $n>1$ is an integer, are defined and ordered as follows. If $a,\,b$ are basic commutators of lengths smaller than $n$, then $[ab]$ is considered to be a basic commutator of length $n$ if and only if the following conditions are met: 1) $a,\,b$ are basic commutators of lengths $k$ and $l$, respectively, and $k+l = n$; 2) $a>b$; and 3) if $a=[cd]$, then $d\le b$. The basic commutators of length not exceeding $n$ thus obtained are arbitrarily ordered, subject to the condition that $[ab] > b$, while preserving the order of the basic commutators of lengths less than $n$. Any set of basic commutators constructed in this way is a base of the free Lie algebra with $R$ as set of free generators [1].

References

 [1] A.I. Shirshov, "On bases of free Lie algebras" Algebra i Logika , 1 : 1 (1962) pp. 14–19 (In Russian)

Let $M(R)$ be the free magma on $R$, i.e. the set of all non-commutative and non-associative words in the alphabet $R$. The basic commutators are to be seen as a subset of $M(R)$. This subset is also often called a P. Hall set. The identity on $R$ induces a mapping $\phi : M(R) \to L_K(R)$, where $L_K(R)$ is the free Lie algebra on $R$ over the ring $K$. Let $K(R)$ be a P. Hall set in $M(R)$ (i.e. a set of basic commutators), then $\phi(H(R))$ is a basis of the free $K$-module $L_K(R)$, called a P. Hall basis. Other useful bases of $L_K(R)$ are the Chen–Fox–Lyndon basis and the Shirshov basis (these two are essentially the same), and the Spitzer–Foata basis; cf. [a4] for these. Let $R$ be finite of cardinality $r = \# R$. Let $l_r(n)$ be the number of basic commutators on $R$ of length $n$. Then
$$l_r(n) = \frac1n \sum_{d|n} \mu(d) r^{n/d}$$
where $\mu : \{1, 2, \ldots\} \to \{-1,0,1\}$ is the Möbius function, defined by $\mu(1) = 1$, $\mu(k)=0$ if $k$ is divisible by a square, and $\mu(p_1\cdots p_m) = (-1)^m$ if $p_1,\ldots,p_m$ are distinct prime numbers.