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''regular 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]]].
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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 [[#References|[1]]].
  
 
====References====
 
====References====
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====Comments====
 
====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
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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. [[#References|[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
  
<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>
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$$
 +
l_r(n) = \frac1n \sum_{d|n} \mu(d) r^{n/d}
 +
$$
  
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.
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where $\mu : \{1, 2, \ldots\} \to \{-1,0,1\}$ is the [[Möbius function|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.
  
 
====References====
 
====References====
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)</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">[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 monoïdes libres" , ''Lect. notes in math.'' , '''691''' , Springer  (1978)</TD></TR></table>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Viennot,  "Algèbres de Lie libres et monoïdes libres" , ''Lect. notes in math.'' , '''691''' , Springer  (1978) {{ZBL|0395.17003}}</TD></TR></table>
  
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Latest revision as of 08:07, 13 February 2024

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)


Comments

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.

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
How to Cite This Entry:
Basic commutator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Basic_commutator&oldid=53587
This article was adapted from an original article by Yu.M. Gorchakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article