Basic commutator
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].
References
[1] | A.I. Shirshov, "On bases of free Lie algebras" Algebra i Logika , 1 : 1 (1962) pp. 14–19 (In Russian) |
Comments
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
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where is the Möbius function, defined by
,
if
is divisible by a square, and
if
are distinct prime numbers.
References
[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 monoïdes libres" , Lect. notes in math. , 691 , Springer (1978) Zbl 0395.17003 |
Basic commutator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Basic_commutator&oldid=54071