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Difference between revisions of "Lazard set"

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m (fixing array)
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A subset  $  T $
 
A subset  $  T $
 
of the [[Free magma|free magma]]  $  M ( A ) $,  
 
of the [[Free magma|free magma]]  $  M ( A ) $,  
i.e. the free non-associative structure over  $  A $(
+
i.e. the free non-associative structure over  $  A $ (cf. also [[Associative rings and algebras|Associative rings and algebras]]). The elements of  $  M ( A ) $
cf. also [[Associative rings and algebras|Associative rings and algebras]]). The elements of  $  M ( A ) $
+
correspond to completely bracketed words over  $  A $ (or rooted planar binary trees with leaves labelled by generators  $  a _ {1} , a _ {2} , \dots $;  
correspond to completely bracketed words over  $  A $(
 
or rooted planar binary trees with leaves labelled by generators  $  a _ {1} , a _ {2} , \dots $;  
 
 
cf. also [[Binary tree|Binary tree]]). These are defined recursively as brackets  $  t = [ t  ^  \prime  , t ^ {\prime \prime } ] $
 
cf. also [[Binary tree|Binary tree]]). These are defined recursively as brackets  $  t = [ t  ^  \prime  , t ^ {\prime \prime } ] $
 
where  $  t  ^  \prime  , t ^ {\prime \prime } $
 
where  $  t  ^  \prime  , t ^ {\prime \prime } $
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$$  
 
$$  
[ \dots [ [ s, t ] , t ] \dots t ]  ( p \textrm{ closing  brackets } ) .
+
[ \dots [ [ s, t ] , t ] \dots t ]  \qquad (p \textrm{ closing  brackets} ) .
 
$$
 
$$
  
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\left .
 
\left .
  
\begin{array}{cttt}
+
\begin{array}{l}
t _ {0} \in T _ {0} = A,  &t _ {1} \in T _ {1} = \left \{ {[ tt _ {0}  ^ {p} ] } : {p \geq  0, t \in T _ {0} \setminus  t _ {0} } \right \} , &\dots  \dots  &t _ {n} \in T _ {n} =  
+
t _ {0} \in T _ {0} = A, \\ t _ {1} \in T _ {1} = \left \{ {[ tt _ {0}  ^ {p} ] } : {p \geq  0, t \in T _ {0} \setminus  t _ {0} } \right \} , \\
 +
  \dots  \dots  \\
 +
t _ {n} \in T _ {n} =  
 
\left \{ {[ tt _ {n - 1 }  ^ {p} ] } : {p \geq  0, t \in T _ {n - 1 }  \setminus  t _ {n - 1 }  } \right \} ,  \\
 
\left \{ {[ tt _ {n - 1 }  ^ {p} ] } : {p \geq  0, t \in T _ {n - 1 }  \setminus  t _ {n - 1 }  } \right \} ,  \\
t _ {1} \in T _ {1} = \left \{ {[ tt _ {0}  ^ {p} ] } : {p \geq  0, t \in T _ {0} \setminus  t _ {0} } \right \} , &\dots  \dots  &t _ {n} \in T _ {n} =  
+
t _ {1} \in T _ {1} = \left \{ {[ tt _ {0}  ^ {p} ] } : {p \geq  0, t \in T _ {0} \setminus  t _ {0} } \right \} , \\
 +
\dots  \dots  \\
 +
t _ {n} \in T _ {n} =  
 
\left \{ {[ tt _ {n - 1 }  ^ {p} ] } : {p \geq  0, t \in T _ {n - 1 }  \setminus  t _ {n - 1 }  } \right \} ,  \\
 
\left \{ {[ tt _ {n - 1 }  ^ {p} ] } : {p \geq  0, t \in T _ {n - 1 }  \setminus  t _ {n - 1 }  } \right \} ,  \\
 
\end{array}
 
\end{array}
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that is, one may associate a [[Lie polynomial|Lie polynomial]]  $  \psi ( t ) $
 
that is, one may associate a [[Lie polynomial|Lie polynomial]]  $  \psi ( t ) $
 
to each element  $  t \in L $
 
to each element  $  t \in L $
of a Lazard set such that the free Lie algebra  $  {\mathcal L} ( A ) $(
+
of a Lazard set such that the free Lie algebra  $  {\mathcal L} ( A ) $ (over  $  A $;  
over  $  A $;  
 
 
cf. [[Lie algebra, free|Lie algebra, free]]) is freely generated (as a module over a commutative ring  $  K $)  
 
cf. [[Lie algebra, free|Lie algebra, free]]) is freely generated (as a module over a commutative ring  $  K $)  
 
by the Lie polynomials  $  \{ {\psi ( t ) } : {t \in L } \} $.  
 
by the Lie polynomials  $  \{ {\psi ( t ) } : {t \in L } \} $.  

Revision as of 06:30, 16 June 2022


A subset $ T $ of the free magma $ M ( A ) $, i.e. the free non-associative structure over $ A $ (cf. also Associative rings and algebras). The elements of $ M ( A ) $ correspond to completely bracketed words over $ A $ (or rooted planar binary trees with leaves labelled by generators $ a _ {1} , a _ {2} , \dots $; cf. also Binary tree). These are defined recursively as brackets $ t = [ t ^ \prime , t ^ {\prime \prime } ] $ where $ t ^ \prime , t ^ {\prime \prime } $ are bracketed words of lower weight; bracketed words of weight one correspond to the generators $ a _ {1} , a _ {2} , \dots $. A subset $ E \subset M ( A ) $ is said to be closed, if for each element $ t = [ t ^ \prime , t ^ {\prime \prime } ] \in E $ one has $ t ^ \prime , t ^ {\prime \prime } \in E $. Given two elements $ s, t \in M ( A ) $, one writes $ [ st ^ {p} ] $ to denote the element

$$ [ \dots [ [ s, t ] , t ] \dots t ] \qquad (p \textrm{ closing brackets} ) . $$

Consider trees $ t _ {0} \dots t _ {n} $ and subsets $ T _ {0} \dots T _ {n + 1 } \subset M ( A ) $ defined as follows:

$$ \tag{a1 } \left . \begin{array}{l} t _ {0} \in T _ {0} = A, \\ t _ {1} \in T _ {1} = \left \{ {[ tt _ {0} ^ {p} ] } : {p \geq 0, t \in T _ {0} \setminus t _ {0} } \right \} , \\ \dots \dots \\ t _ {n} \in T _ {n} = \left \{ {[ tt _ {n - 1 } ^ {p} ] } : {p \geq 0, t \in T _ {n - 1 } \setminus t _ {n - 1 } } \right \} , \\ t _ {1} \in T _ {1} = \left \{ {[ tt _ {0} ^ {p} ] } : {p \geq 0, t \in T _ {0} \setminus t _ {0} } \right \} , \\ \dots \dots \\ t _ {n} \in T _ {n} = \left \{ {[ tt _ {n - 1 } ^ {p} ] } : {p \geq 0, t \in T _ {n - 1 } \setminus t _ {n - 1 } } \right \} , \\ \end{array} \right \} $$

A Lazard set is a subset $ L \subset M ( A ) $ such that for any finite, non-empty and closed subset $ E \subset M ( A ) $ one has:

$$ L \cap E = \{ t _ {0} > \dots > t _ {n} \} $$

for some $ n \geq 0 $, (a1) holds and, moreover, $ T _ {n + 1 } \cap E = \emptyset $.

Lazard sets may be shown to coincide with Hall sets (cf. Hall set). Thus, they give bases of the free Lie algebra over $ A $; that is, one may associate a Lie polynomial $ \psi ( t ) $ to each element $ t \in L $ of a Lazard set such that the free Lie algebra $ {\mathcal L} ( A ) $ (over $ A $; cf. Lie algebra, free) is freely generated (as a module over a commutative ring $ K $) by the Lie polynomials $ \{ {\psi ( t ) } : {t \in L } \} $. Lazard's elimination process may then be phrased as follows: One has the direct sum decomposition (as a module over a commutative ring $ K $):

$$ {\mathcal L} ( A ) = K \psi ( t _ {0} ) \oplus \dots \oplus K \psi ( t _ {n} ) \oplus {\mathcal L} _ {n + 1 } , $$

where $ {\mathcal L} _ {n + 1 } $ is the Lie subalgebra freely generated by $ T _ {n + 1 } $.

Lazard sets were introduced by X. Viennot [a1] in order to unify combinatorial constructions of bases of the free Lie algebra. The Lyndon basis (see Lyndon word) was thought to be of a different nature from the one considered by M. Hall [a2], and generalizations of it were proposed by many authors. Viennot gave a unifying framework for all these constructions. One may present Lazard sets in terms of words, rather than trees in $ M ( A ) $. It can then be shown that a unique tree structure is attached to every word of a Lazard set. Moreover, a Lazard set of words is totally ordered, as is a Lazard set of trees, and it is a complete factorization of the free monoid. That is, every word is a unique non-increasing product of Lazard words. This result makes explicit the link between bases of free Lie algebras and complete factorizations of free monoids.

See also Hall word.

References

[a1] X. Viennot, "Algèbres de Lie libres et monoïdes libres" , Lecture Notes in Mathematics , 691 , Springer (1978)
[a2] M. Hall, "A basis for free Lie rings and higher commutators in free groups" Proc. Amer. Math. Soc. , 1 (1950) pp. 57–581
[a3] C. Reutenauer, "Free Lie algebras" , London Math. Soc. Monographs New Ser. , 7 , Oxford Univ. Press (1993)
How to Cite This Entry:
Lazard set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lazard_set&oldid=52443
This article was adapted from an original article by G. Melançon (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article