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| In [[mathematics]], a '''free Lie algebra''', over a given [[field (mathematics)|field]] ''K'', is a [[Lie algebra]] generated by a set ''X'', without any imposed relations.
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| ==Definition==
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| [[Image:Free lie.png|right|thumb|100px]]
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| : Let ''X'' be a set and ''i'': ''X'' → ''L'' a morphism of sets from ''X'' into a Lie algebra ''L''. The Lie algebra ''L'' is called '''free on ''X''''' if for any Lie algebra ''A'' with a morphism of sets ''f'': ''X'' → ''A'', there is a unique Lie algebra morphism ''g'': ''L'' → ''A'' such that ''f'' = ''g'' o ''i''.
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| Given a set ''X'', one can show that there exists a unique free Lie algebra ''L(X)'' generated by ''X''.
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| In the language of category theory, the [[functor]] sending a set ''X'' to the Lie algebra generated by ''X'' is the [[Free_object#Free_functor|free functor]] from the category of sets to the category of Lie algebras. That is, it is [[left adjoint]] to the [[forgetful functor]].
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| As the 0-graded component of the free Lie algebra on a set ''X'' is just the free vector space on that group, one can alternatively define a free Lie algebra on a vector space ''V'' as left adjoint to the forgetful functor from Lie algebras over a field ''K'' to vector spaces over the field ''K'' – forgetting the Lie algebra structure, but remembering the vector space structure.
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| ==Universal enveloping algebra==
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| The [[universal enveloping algebra]] of a free Lie algebra on a set ''X'' is the [[free associative algebra]] generated by ''X''. By the [[Poincaré-Birkhoff-Witt theorem]] it is the "same size" as the symmetric algebra of the free Lie algebra (meaning that if both sides are graded by giving elements of ''X'' degree 1 then they are isomorphic as graded vector spaces). This can be used to describe the dimension of the piece of the free Lie algebra of any given degree.
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| Witt showed that the number of basic commutators of degree ''k'' in the free Lie algebra on an ''m''-element set is given by the [[necklace polynomial]]:
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| :<math>N_k = \frac{1}{k}\sum_{d|k}\mu(d)\cdot m^{k/d},</math>
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| where <math>\mu</math> is the [[Möbius function]].
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| The graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the [[shuffle algebra]].
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| ==Hall sets==
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| An explicit basis of the free Lie algebra can be given in terms of a '''Hall set''', which is a particular kind of subset inside the [[free magma]] on ''X''. Elements of the free magma are [[binary tree]]s, with their leaves labelled by elements of ''X''. Hall sets were introduced by {{harvs|txt||first=Marshall |last=Hall|authorlink=Marshall Hall (mathematician)|year=1950}} based on work of [[Philip Hall]] on groups. Subsequently [[Wilhelm Magnus]] showed that they arise as the [[graded Lie algebra]] associated with the filtration on a [[free group]] given by the [[lower central series]]. This correspondence was motivated by [[commutator]] identities in [[group theory]] due to Philip Hall and [[Ernst Witt]].
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| ==Lyndon basis==
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| In particular there is a basis of the free Lie algebra corresponding to [[Lyndon word]]s, called the '''Lyndon basis'''. (This is also called the Chen–Fox–Lyndon basis or the Lyndon–Shirshov basis, and is essentially the same as the '''Shirshov basis'''.)
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| There is a bijection γ from the Lyndon words in an ordered alphabet to a basis of the free Lie algebra on this alphabet defined as follows.
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| *If a word ''w'' has length 1 then γ(''w'')=''w'' (considered as a generator of the free Lie algebra).
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| *If ''w'' has length at least 2, then write ''w''=''uv'' for Lyndon words ''u'', ''v'' with ''v'' as long as possible. Then γ(''w'') = [γ(''u''),γ(''v'')]
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| == Shirshov–Witt theorem ==
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| {{harvs|txt|last=Širšov|authorlink=Anatoly Illarionovich Shirshov|year=1953}} and {{harvs|txt|last=Witt|year=1956}} showed that any [[Lie subalgebra]] of a free Lie algebra is itself a free Lie algebra.
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| ==Applications==
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| The Milnor invariants of the [[link group]] are related to the free Lie algebra, as discussed in that article.
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| ==See also==
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| *[[Free object]]
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| *[[Free algebra]]
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| *[[Free group]]
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| ==References==
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| {{reflist}}
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| * {{springer|id=l/l058410|title=Free Lie algebra over a ring|first=Yu.A. |last=Bakhturin}}
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| * N. Bourbaki, "Lie Groups and Lie Algebras," Chapter II: Free Lie Algebras, Springer, 1989. ISBN 0-387-50218-1
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| *{{Citation | last1=Chen | first1=Kuo-Tsai | not-used-author1-link=Kuo-Tsai Chen | last2=Fox | first2=Ralph H. | author2-link=Ralph Fox | last3=Lyndon | first3=Roger C. | author3-link=Roger Lyndon | title=Free differential calculus. IV. The quotient groups of the lower central series | jstor=1970044 | mr=0102539 | year=1958 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=68 | pages=81–95 | issue=1 | doi=10.2307/1970044}}
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| *{{Citation | last1=Hall | first1=Marshall | title=A basis for free Lie rings and higher commutators in free groups | url=http://www.ams.org/journals/proc/1950-001-05/S0002-9939-1950-0038336-7/ | doi=10.1090/S0002-9939-1950-0038336-7 | mr=0038336 | year=1950 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=1 | pages=575–581 | issue=5}}
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| *{{Citation | last=Lothaire | first=M. | authorlink=M. Lothaire | others=Perrin, D.; Reutenauer, C.; Berstel, J.; Pin, J. E.; Pirillo, G.; Foata, D.; Sakarovitch, J.; Simon, I.; Schützenberger, M. P.; Choffrut, C.; Cori, R.; Lyndon, Roger; Rota, Gian-Carlo. Foreword by Roger Lyndon | title=Combinatorics on words | edition=2nd | series=Encyclopedia of Mathematics and Its Applications | volume=17 | publisher=[[Cambridge University Press]] | year=1997 | isbn=0-521-59924-5 | zbl=0874.20040 | pages=76-91,98 }}
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| *{{Citation | last1=Magnus | first1=Wilhelm | author1-link=Wilhelm Magnus | title=Über Beziehungen zwischen höheren Kommutatoren | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN00217412X | language=German | doi=10.1515/crll.1937.177.105 | jfm=63.0065.01 | year=1937 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=177 | pages=105–115 | issue=177}}
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| * W. Magnus, A. Karrass, D. Solitar, "Combinatorial group theory". Reprint of the 1976 second edition, Dover, 2004. ISBN 0-486-43830-9
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| * {{springer|id=h/h110040|title=Hall set|author=G. Melançon}}
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| * {{springer|id=h/h110050|title=Hall word|author=G. Melançon}}
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| *{{eom|id=/S/s110100|first=G. |last=Melançon|title=Shirshov basis}}
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| *{{Citation | last1=Reutenauer | first1=Christophe | title=Free Lie algebras | url=http://books.google.com/books?id=cBvvAAAAMAAJ | publisher=The Clarendon Press Oxford University Press | series=London Mathematical Society Monographs. New Series | isbn=978-0-19-853679-6 | mr=1231799 | year=1993 | volume=7}}
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| *{{Citation | last1=Širšov | first1=A. I. | title=Subalgebras of free Lie algebras | mr=0059892 | year=1953 | journal=Mat. Sbornik N.S. | volume=33(75) | pages=441–452}}
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| *{{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=Die Unterringe der freien Lieschen Ringe | doi=10.1007/BF01166568 | mr=0077525 | year=1956 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=64 | pages=195–216}}
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| [[Category:Properties of Lie algebras]]
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| [[Category:Free algebraic structures]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. I've always cherished residing in Alaska. I am really fond of handwriting but I can't make it my occupation really. Invoicing is what I do for a residing but I've always needed my own business.
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