Hyperbolic group - Revision history

en>Monkbot: /* References */Task 3: Fix CS1 deprecated coauthor parameter errors

2014-05-12T04:30:45Z

References: Task 3: Fix CS1 deprecated coauthor parameter errors

← Older revision		Revision as of 06:30, 12 May 2014
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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~~.<br><br>my ~~blog post: email~~ psychic ~~readings~~ ([http://~~www~~.~~indosfriends~~.~~com~~/~~profile-253/info/ indosfriends~~.~~com~~])		The writer's name is Christy Brookins. Office supervising is where her primary income arrives from. I've usually cherished residing in Alaska. What I adore performing is football but I don't have the time lately.<br><br>Here is my web site; phone psychic ([http://chorokdeul.co.kr/index.php?document_srl=324263&mid=customer21 chorokdeul.co.kr])

en>BD2412: /* Properties */Fixing links to disambiguation pages using AWB

2014-02-20T20:28:38Z

Properties: Fixing links to disambiguation pages using AWB

← Older revision		Revision as of 22:28, 20 February 2014
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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~~.		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.<br><br>my blog post: email psychic readings ([http://www.indosfriends.com/profile-253/info/ indosfriends.com])

	~~==Definition==~~
	~~[[Image:Free lie.png\|right\|thumb\|100px]]~~
	~~: 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''.~~
	~~Given a set ''X'', one~~ can ~~show that there exists a unique free Lie algebra ''L(X)'' generated by ''X''.~~

	~~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]]~~.

	~~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.~~

	~~==Universal enveloping algebra==~~

	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~~.

	~~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]]:~~
	:<~~math~~>~~N_k = \frac{1}{k}\sum_{d\|k}\mu(d)\cdot m^{k/d},~~<~~/math~~>
	~~where <math>\mu</math> is the [[Möbius function]].~~

	~~The graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the [[shuffle algebra]].~~

	~~==Hall sets==~~

	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]].

	~~==Lyndon basis==~~

	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'''.)
	~~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.~~
	*If a word ''w'' has length 1 then γ(''w'')=''w'' (~~considered as a generator of the free Lie algebra).~~
	*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'')]

	~~== Shirshov–Witt theorem ==~~

	{{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.

	~~==Applications==~~
	~~The Milnor invariants of the [[link group]] are related to the free Lie algebra, as discussed in that article.~~

	~~==See also==~~
	*[[Free object]]
	*[[Free algebra]]
	*[[Free group]]

	~~==References==~~
	~~{{reflist}}~~
	* {{springer\|id=l/l058410\|title=Free Lie algebra over a ring\|first=Yu.A. \|last=Bakhturin}}
	* N. Bourbaki, "Lie Groups and Lie Algebras," Chapter II: Free Lie Algebras, Springer, 1989. ISBN 0-387-50218-1
	*{{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}}~~
	*{{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}}~~
	*{{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 }}~~
	*{{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}}
	* W. Magnus, A. Karrass, D. Solitar, "Combinatorial group theory". Reprint of the 1976 second edition, Dover, 2004. ISBN 0-486-43830-9
	* {{springer\|id=h/h110040\|title=Hall set\|author=G. Melançon}}
	* {{springer\|id=h/h110050\|title=Hall word\|author=G. Melançon}}
	*{{eom\|id=/S/s110100\|first=G. \|last=Melançon\|title=Shirshov basis}}
	*{{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}}
	*{{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}}~~
	*{{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}}

	~~[[Category:Properties of Lie algebras]]~~
	~~[[Category:Free algebraic structures]]~~

en>Yobot: /* Properties */WP:CHECKWIKI errors fixed + general fixes using AWB (8961)

2013-03-07T23:55:15Z

Properties: WP:CHECKWIKI errors fixed + general fixes using AWB (8961)

← Older revision		Revision as of 01:55, 8 March 2013
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	The ~~person who wrote~~ the ~~post~~ is ~~known~~ as ~~Jayson Hirano and he completely digs that name~~. ~~Alaska~~ is ~~exactly where he~~'~~s always been living~~. ~~Office supervising~~ is ~~my profession~~. ~~Doing ballet~~ is ~~something she would~~ by ~~no means give up.~~<br><br>~~my homepage~~ :: ~~online psychic chat (~~[http://~~koreanyelp~~.com/~~index~~.~~php?document_srl~~=~~1798&mid~~=~~SchoolNews linked internet page~~])		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.

			==Definition==
			[[Image:Free lie.png\|right\|thumb\|100px]]
			: 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''.
			Given a set ''X'', one can show that there exists a unique free Lie algebra ''L(X)'' generated by ''X''.

			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]].

			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.

			==Universal enveloping algebra==

			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.

			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]]:
			:<math>N_k = \frac{1}{k}\sum_{d\|k}\mu(d)\cdot m^{k/d},</math>
			where <math>\mu</math> is the [[Möbius function]].

			The graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the [[shuffle algebra]].

			==Hall sets==

			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]].

			==Lyndon basis==

			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'''.)
			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.
			*If a word ''w'' has length 1 then γ(''w'')=''w'' (considered as a generator of the free Lie algebra).
			*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'')]

			== Shirshov–Witt theorem ==

			{{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.

			==Applications==
			The Milnor invariants of the [[link group]] are related to the free Lie algebra, as discussed in that article.

			==See also==
			*[[Free object]]
			*[[Free algebra]]
			*[[Free group]]

			==References==
			{{reflist}}
			* {{springer\|id=l/l058410\|title=Free Lie algebra over a ring\|first=Yu.A. \|last=Bakhturin}}
			* N. Bourbaki, "Lie Groups and Lie Algebras," Chapter II: Free Lie Algebras, Springer, 1989. ISBN 0-387-50218-1
			*{{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}}
			*{{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}}
			*{{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 }}
			*{{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}}
			* W. Magnus, A. Karrass, D. Solitar, "Combinatorial group theory". Reprint of the 1976 second edition, Dover, 2004. ISBN 0-486-43830-9
			* {{springer\|id=h/h110040\|title=Hall set\|author=G. Melançon}}
			* {{springer\|id=h/h110050\|title=Hall word\|author=G. Melançon}}
			*{{eom\|id=/S/s110100\|first=G. \|last=Melançon\|title=Shirshov basis}}
			*{{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}}
			*{{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}}
			*{{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}}

			[[Category:Properties of Lie algebras]]
			[[Category:Free algebraic structures]]

en>ZéroBot: r2.7.1) (Robot: Adding de:Hyperbolische Gruppe

2012-08-29T14:03:00Z

r2.7.1) (Robot: Adding de:Hyperbolische Gruppe

New page

The person who wrote the post is known as Jayson Hirano and he completely digs that name. Alaska is exactly where he's always been living. Office supervising is my profession. Doing ballet is something she would by no means give up.<br><br>my homepage :: online psychic chat ([http://koreanyelp.com/index.php?document_srl=1798&mid=SchoolNews linked internet page])