Free Lie algebra

<h2 id="definition">Definition</h2>

The definition of the free Lie algebra generated by a set X is as follows:

Let X be a set and 
 
 
 
 i
 :
 X
 →
 L
 
 
 {\displaystyle i\colon X\to L}
 
 a <a href="/facts/Morphism/Kdpuyz3S">morphism</a> of sets (<a href="/facts/Function_(mathematics)/CdReEKCh">function</a>) from X into a Lie algebra L. The Lie algebra L is called free on X if 
 
 
 
 i
 
 
 {\displaystyle i}
 
 is the <a href="/facts/Universal_morphism/5XJBYMNz">universal morphism</a>; that is, if for any Lie algebra A with a morphism of sets 
 
 
 
 f
 :
 X
 →
 A
 
 
 {\displaystyle f\colon X\to A}
 
, there is a unique Lie algebra morphism 
 
 
 
 g
 :
 L
 →
 A
 
 
 {\displaystyle g\colon L\to A}
 
 such that 
 
 
 
 f
 =
 g
 ∘
 i
 
 
 {\displaystyle f=g\circ i}
 
.
Given a set X, one can show that there exists a unique free Lie algebra 
 
 
 
 L
 (
 X
 )
 
 
 {\displaystyle L(X)}
 
 generated by X.
In the language of <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, the <a href="/facts/Functor/l2u3rIBE">functor</a> sending a set X to the Lie algebra generated by X is the <a href="/facts/Free_object/VxL28y8B">free functor</a> from the <a href="/facts/Category_of_sets/LF5ykYl0">category of sets</a> to the category of Lie algebras. That is, it is <a href="/facts/Left_adjoint/bJ1P7AN2">left adjoint</a> to the <a href="/facts/Forgetful_functor/onEwu6HA">forgetful functor</a>.
The free Lie algebra on a set X is naturally <a href="/facts/Graded_Lie_algebra/AP0JJLCX">graded</a>. The 1-graded component of the free Lie algebra is just the <a href="/facts/Free_vector_space/o6rC6yoJ">free vector space</a> on that set.
One can alternatively define a free Lie algebra on a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 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.

<h2 id="universal-enveloping-algebra">Universal enveloping algebra</h2>
The <a href="/facts/Universal_enveloping_algebra/VJ3Syz4W">universal enveloping algebra</a> of a free Lie algebra on a set X is the <a href="/facts/Free_associative_algebra/xMrT0WbT">free associative algebra</a> generated by X. By the <a href="/facts/Poincar%25C3%25A9%25E2%2580%2593Birkhoff%25E2%2580%2593Witt_theorem/YSGbvgQv">Poincaré–Birkhoff–Witt theorem</a> 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 <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> as graded vector spaces). This can be used to describe the dimension of the piece of the free Lie algebra of any given degree.
<a href="/facts/Ernst_Witt/w4RfUYpv">Ernst Witt</a> showed that the number of <a href="/facts/Basic_commutator/f1IefaGS">basic commutators</a> of degree k in the free Lie algebra on an m-element set is given by the <a href="/facts/Necklace_polynomial/x3J94taX">necklace polynomial</a>:

M
          
            m
          
        
        (
        k
        )
        =
        
          
            1
            k
          
        
        
          ∑
          
            d
            
              |
            
            k
          
        
        μ
        (
        d
        )
        ⋅
        
          m
          
            k
            
              /
            
            d
          
        
        ,
      
    
    {\displaystyle M_{m}(k)={\frac {1}{k}}\sum _{d|k}\mu (d)\cdot m^{k/d},}

where 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is the <a href="/facts/M%25C3%25B6bius_function/geuvVfqS">Möbius function</a>.
The graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the <a href="/facts/Shuffle_algebra/AgsGQqgJ">shuffle algebra</a>. This essentially follows because universal enveloping algebras have the structure of a <a href="/facts/Hopf_algebra/3QF6g0tT">Hopf algebra</a>, and the <a href="/facts/Shuffle_product/AgsGQqgJ">shuffle product</a> describes the action of comultiplication in this algebra. See <a href="/facts/Tensor_algebra/H7CKKryf">tensor algebra</a> for a detailed exposition of the inter-relation between the shuffle product and comultiplication.

<h2 id="hall-sets">Hall sets</h2>
Main article: <a href="/facts/Hall_set/twm046uI">Hall set</a>
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 <a href="/facts/Free_magma/9g3vp7Wl">free magma</a> on X. Elements of the free magma are <a href="/facts/Binary_tree/TDV7GMpu">binary trees</a>, with their leaves labelled by elements of X. Hall sets were introduced by <a href="/facts/Marshall_Hall_(mathematician)/wLpjsYnL">Marshall Hall</a> (1950) based on work of <a href="/facts/Philip_Hall/4Gal202l">Philip Hall</a> on groups. Subsequently, <a href="/facts/Wilhelm_Magnus/WhmRYJL1">Wilhelm Magnus</a> showed that they arise as the <a href="/facts/Graded_Lie_algebra/AP0JJLCX">graded Lie algebra</a> associated with the filtration on a <a href="/facts/Free_group/zPo2NIGb">free group</a> given by the <a href="/facts/Lower_central_series/Hw9lbKGf">lower central series</a>. This correspondence was motivated by <a href="/facts/Commutator/eYxzFmFY">commutator</a> identities in <a href="/facts/Group_theory/NfowcI5H">group theory</a> due to Philip Hall and Witt.

<h2 id="lyndon-basis">Lyndon basis</h2>
The <a href="/facts/Lyndon_word/ckrmLgXQ">Lyndon words</a> are a special case of the <a href="/facts/Hall_word/twm046uI">Hall words</a>, and so in particular there is a basis of the free Lie algebra corresponding to Lyndon words. This is called the Lyndon basis, named after <a href="/facts/Roger_Lyndon/ESCf6Rr3">Roger Lyndon</a>. (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 <a href="/facts/Bijection/j4gTuTmW">bijection</a> γ from the Lyndon words in an ordered alphabet to a basis of the free Lie algebra on this alphabet defined as follows:

<ul><li>If a word w has length 1 then 
 
 
 
 γ
 (
 w
 )
 =
 w
 
 
 {\displaystyle \gamma (w)=w}
 
 (considered as a generator of the free Lie algebra).</li>
<li>If w has length at least 2, then write 
 
 
 
 w
 =
 u
 v
 
 
 {\displaystyle w=uv}
 
 for Lyndon words u, v with v as long as possible (the "standard factorization"<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>). Then 
 
 
 
 γ
 (
 w
 )
 =
 [
 γ
 (
 u
 )
 ,
 γ
 (
 v
 )
 ]
 
 
 {\displaystyle \gamma (w)=[\gamma (u),\gamma (v)]}
 
.</li></ul>
<h2 id="shirshovwitt-theorem">Shirshov–Witt theorem</h2>
<a href="/facts/Anatoly_Shirshov/76KreJCj">Anatoly Širšov</a> (1953) and <a href="/facts/Ernst_Witt/w4RfUYpv">Witt</a> (1956) showed that any <a href="/facts/Lie_algebra/n2gAUkNq">Lie subalgebra</a> of a free Lie algebra is itself a free Lie algebra.

<h2 id="applications">Applications</h2>
<a href="/facts/Serre%2527s_theorem_on_a_semisimple_Lie_algebra/KPM3h8It">Serre's theorem on a semisimple Lie algebra</a> uses a free Lie algebra to construct a semisimple algebra out of generators and relations.
The <a href="/facts/Link_group/IpJxERVo">Milnor invariants</a> of a <a href="/facts/Link_group/IpJxERVo">link group</a> are related to the free Lie algebra on the components of the <a href="/facts/Link_(knot_theory)/hoFDYmVj">link</a>, as discussed in that article.
See also <a href="/facts/Lie_operad/QchLFYR9">Lie operad</a> for the use of a free Lie algebra in the construction of the operad.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Free_object/VxL28y8B">Free object</a></li>
<li><a href="/facts/Free_algebra/xMrT0WbT">Free algebra</a></li>
<li><a href="/facts/Free_group/zPo2NIGb">Free group</a></li></ul>

<ul><li>Bakhturin, Yu.A. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Free_Lie_algebra_over_a_ring">"Free Lie algebra over a ring"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li><a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki, Nicolas</a> (1989). "Chapter II: Free Lie Algebras". Lie Groups and Lie Algebras. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-50218-1.</li>
<li>Chen, Kuo-Tsai; <a href="/facts/Ralph_Fox/GFlG3J5v">Fox, Ralph H.</a>; <a href="/facts/Roger_Lyndon/ESCf6Rr3">Lyndon, Roger C.</a> (1958), "Free differential calculus. IV. The quotient groups of the lower central series", <a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a>, Second Series, 68 (1): 81–95, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1970044">10.2307/1970044</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1970044">1970044</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0102539">0102539</a></li>
<li><a href="/facts/Marshall_Hall_(mathematician)/wLpjsYnL">Hall, Marshall</a> (1950), <a href="https://www.ams.org/journals/proc/1950-001-05/S0002-9939-1950-0038336-7/">"A basis for free Lie rings and higher commutators in free groups"</a>, <a href="/facts/Proceedings_of_the_American_Mathematical_Society/W3wRF3MV">Proceedings of the American Mathematical Society</a>, 1 (5): 575–581, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0002-9939-1950-0038336-7">10.1090/S0002-9939-1950-0038336-7</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9939">0002-9939</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0038336">0038336</a></li>
<li><a href="/facts/M._Lothaire/pC43ftpJ">Lothaire, M.</a> (1997), Combinatorics on words, Encyclopedia of Mathematics and Its Applications, vol. 17, Perrin, D.; Reutenauer, Christophe; Berstel, J.; Pin, J. E.; Pirillo, G.; Foata, D.; Sakarovitch, J.; Simon, I.; <a href="/facts/Marcel-Paul_Sch%25C3%25BCtzenberger/aF62Lo9H">Schützenberger, Marcel-Paul</a>; Choffrut, C.; Cori, R.; <a href="/facts/Roger_Lyndon/ESCf6Rr3">Lyndon, Roger</a>; <a href="/facts/Gian-Carlo_Rota/zLSPLVwR">Rota, Gian-Carlo</a>. Foreword by Roger Lyndon (2nd ed.), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, pp. 76–91, 98, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-59924-5, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0874.20040">0874.20040</a></li>
<li><a href="/facts/Wilhelm_Magnus/WhmRYJL1">Magnus, Wilhelm</a> (1937), <a href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN00217412X">"Über Beziehungen zwischen höheren Kommutatoren"</a>, <a href="/facts/Crelle%2527s_Journal/YCcRSaAQ">Journal für die Reine und Angewandte Mathematik</a> (in German), 1937 (177): 105–115, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2Fcrll.1937.177.105">10.1515/crll.1937.177.105</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>, <a href="/facts/JFM_(identifier)/P6rFxKKx">JFM</a> <a href="https://zbmath.org/?format=complete&q=an:63.0065.01">63.0065.01</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:199546158">199546158</a></li>
<li><a href="/facts/Wilhelm_Magnus/WhmRYJL1">Magnus, Wilhelm</a>; Karrass, Abraham; <a href="/facts/Donald_Solitar/0tIqNUSa">Solitar, Donald</a> (2004). Combinatorial group theory (Reprint of the 1976 second ed.). Mineola, NY: <a href="/facts/Dover_Publications/RwjvlSFv">Dover</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-43830-9. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2109550">2109550</a>.</li>
<li>Guy Melançon (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Hall_set">"Hall set"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Guy Melançon (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Hall_word">"Hall word"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Melançon, Guy (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Shirshov_basis">"Shirshov basis"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Reutenauer, Christophe (1993), <a href="https://books.google.com/books?id=cBvvAAAAMAAJ">Free Lie algebras</a>, London Mathematical Society Monographs. New Series, vol. 7, The Clarendon Press <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-853679-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1231799">1231799</a></li>
<li>Širšov, Anatoliĭ I. (1953), "Subalgebras of free Lie algebras", Mat. Sbornik, New Series, 33 (75): 441–452, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0059892">0059892</a></li>
<li><a href="/facts/Anatoly_Shirshov/76KreJCj">Širšov, Anatoliĭ I.</a> (1958), "On free Lie rings", Mat. Sbornik, New Series, 45 (2): 113–122, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0099356">0099356</a></li>
<li>Bokut, Leonid A.; Latyshev, Victor; Shestakov, Ivan; <a href="/facts/Efim_Zelmanov/14wssqSY">Zelmanov, Efim</a>, eds. (2009). Selected works of A.I. Shirshov. Translated by Bremner, Murray; Kochetov, Mikhail V. Basel, Boston, Berlin: Birkhäuser. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2547481">2547481</a>.</li>
<li><a href="/facts/Ernst_Witt/w4RfUYpv">Witt, Ernst</a> (1956). "Die Unterringe der freien Lieschen Ringe". <a href="/facts/Mathematische_Zeitschrift/8JqT7561">Mathematische Zeitschrift</a>. 64: 195–216. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01166568">10.1007/BF01166568</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0077525">0077525</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119607181">119607181</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Berstel, Jean; Perrin, Dominique (2007), "The origins of combinatorics on words" (PDF), European Journal of Combinatorics, 28 (3): 996–1022, doi:10.1016/j.ejc.2005.07.019, MR 2300777 <a href="http://www-igm.univ-mlv.fr/~berstel/Articles/2007Origins.pdf" target="_blank">http://www-igm.univ-mlv.fr/~berstel/Articles/2007Origins.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Free Lie algebra open-in-new

Free Lie algebra