View source for Basic commutator

''regular commutator''

An object inductively constructed from the elements of a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b0153301.png" /> and from brackets, in the following manner. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b0153302.png" /> are considered by definition to be basic commutators of length 1, and they are given an arbitrary total order. The basic commutators of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b0153303.png" />, where where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b0153304.png" /> is an integer, are defined and ordered as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b0153305.png" /> are basic commutators of lengths smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b0153306.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b0153307.png" /> is considered to be a basic commutator of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b0153308.png" /> if and only if the following conditions are met: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b0153309.png" /> are basic commutators of lengths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533011.png" />, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533012.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533013.png" />; and 3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533015.png" />. The basic commutators of length not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533016.png" /> thus obtained are arbitrarily ordered, subject to the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533017.png" />, while preserving the order of the basic commutators of lengths less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533018.png" />. Any set of basic commutators constructed in this way is a base of the free Lie algebra with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533019.png" /> as set of free generators [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Shirshov,   "On bases of free Lie algebras"  ''Algebra i Logika'' , '''1''' :  1  (1962)  pp. 14–19  (In Russian)</TD></TR></table>



====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533020.png" /> be the free magma on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533021.png" />, i.e. the set of all non-commutative and non-associative words in the alphabet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533022.png" />. The basic commutators are to be seen as a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533023.png" />. This subset is also often called a P. Hall set. The identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533024.png" /> induces a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533026.png" /> is the free Lie algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533027.png" /> over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533028.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533029.png" /> be a P. Hall set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533030.png" /> (i.e. a set of basic commutators), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533031.png" /> is a basis of the free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533032.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533033.png" />, called a P. Hall basis. Other useful bases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533034.png" /> are the Chen–Fox–Lyndon basis and the Shirshov basis (these two are essentially the same), and the Spitzer–Foata basis; cf. [[#References|[a4]]] for these. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533035.png" /> be finite of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533036.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533037.png" /> be the number of basic commutators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533038.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533039.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533040.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533041.png" /> is the [[Möbius function|Möbius function]], defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533043.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533044.png" /> is divisible by a square, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533045.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015330/b01533046.png" /> are distinct prime numbers.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,   "Groupes et algèbres de Lie" , Hermann  (1972)  pp. Chapt. 2; 3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hall jr.,   "The theory of groups" , Macmillan  (1959)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Magnus,   A. Karrass,   B. Solitar,   "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)  pp. 412</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Viennot,   "Algèbres de Lie libres et monoides libres" , ''Lect. notes in math.'' , '''691''' , Springer  (1978)</TD></TR></table>

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