Basic commutator

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regular commutator

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


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


Let be the free magma on , i.e. the set of all non-commutative and non-associative words in the alphabet . The basic commutators are to be seen as a subset of . This subset is also often called a P. Hall set. The identity on induces a mapping , where is the free Lie algebra on over the ring . Let be a P. Hall set in (i.e. a set of basic commutators), then is a basis of the free -module , called a P. Hall basis. Other useful bases of 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 be finite of cardinality . Let be the number of basic commutators on of length . Then

where is the Möbius function, defined by , if is divisible by a square, and if are distinct prime numbers.


[a1] N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1972) pp. Chapt. 2; 3
[a2] M. Hall jr., "The theory of groups" , Macmillan (1959)
[a3] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) pp. 412
[a4] G. Viennot, "Algèbres de Lie libres et monoides libres" , Lect. notes in math. , 691 , Springer (1978)
How to Cite This Entry:
Basic commutator. Yu.M. Gorchakov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098