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''over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l0584101.png" />''
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''over a ring $R$''
  
A [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l0584102.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l0584103.png" /> in which one can distinguish a free generating set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l0584104.png" />, a mapping from which into an arbitrary algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l0584105.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l0584106.png" /> can be be extended to a homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l0584107.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l0584108.png" />. The cardinality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l0584109.png" /> completely determines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841010.png" /> and is called its rank. A free Lie algebra is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841011.png" />-module (for bases of it see [[Basic commutator|Basic commutator]]). A subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841012.png" /> of a free Lie algebra over a field is itself a free Lie algebra (Shirshov's theorem, [[#References|[1]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841013.png" />, then this is true only under the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841014.png" /> is a free Abelian group [[#References|[2]]]. The finitely-generated subalgebras of a free Lie algebra over a field form a sublattice of the lattice of all subalgebras [[#References|[3]]]. W. Magnus [[#References|[4]]] established canonical connections between free Lie algebras and both free groups and free associative algebras.
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A [[Lie algebra]] $L = L(X)$ over $R$ in which one can distinguish a free generating set $X$, a mapping from which into an arbitrary algebra $G$ over $R$ can be be extended to a homomorphism from $L$ into $G$. The cardinality of $X$ completely determines $L$ and is called its rank. A free Lie algebra is a free $R$-module (for bases of it see [[Basic commutator]]). A subalgebra $M$ of a free Lie algebra over a field is itself a free Lie algebra (Shirshov's theorem, [[#References|[1]]]). If $R = \mathbf{Z}$, then this is true only under the condition that $L/M$ is a free Abelian group [[#References|[2]]]. The finitely-generated subalgebras of a free Lie algebra over a field form a sublattice of the lattice of all subalgebras [[#References|[3]]]. W. Magnus [[#References|[4]]] established canonical connections between free Lie algebras and both free groups and free associative algebras.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Shirshov,  "Subalgebras of free Lie algebras"  ''Mat. Sb.'' , '''33''' :  2  (1953)  pp. 441–452  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Witt,  "Die Unterringe der freien Lieschen Ringe"  ''Math. Z.'' , '''64'''  (1956)  pp. 195–216</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.P. Kukin,  "Intersection of subalgebras of a free Lie algebra"  ''Algebra and Logic'' , '''16'''  (1977)  pp. 387–394  ''Algebra i Logika'' , '''16'''  (1977)  pp. 577–587</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Magnus,  "Ueber Beziehungen zwischen höheren Kommutatoren"  ''J. Reine Angew. Math.'' , '''177'''  (1937)  pp. 105–115</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Yu.A. Bakhturin,  "Identical relations in Lie algebras" , VNU , Utrecht  (1987)  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Shirshov,  "Subalgebras of free Lie algebras"  ''Mat. Sb.'' , '''33''' :  2  (1953)  pp. 441–452  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  E. Witt,  "Die Unterringe der freien Lieschen Ringe"  ''Math. Z.'' , '''64'''  (1956)  pp. 195–216</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  G.P. Kukin,  "Intersection of subalgebras of a free Lie algebra"  ''Algebra and Logic'' , '''16'''  (1977)  pp. 387–394  ''Algebra i Logika'' , '''16'''  (1977)  pp. 577–587</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  W. Magnus,  "Ueber Beziehungen zwischen höheren Kommutatoren"  ''J. Reine Angew. Math.'' , '''177'''  (1937)  pp. 105–115</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top">  Yu.A. Bakhturin,  "Identical relations in Lie algebras" , VNU , Utrecht  (1987)  (Translated from Russian)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|part}}

Revision as of 16:13, 23 April 2016

over a ring $R$

A Lie algebra $L = L(X)$ over $R$ in which one can distinguish a free generating set $X$, a mapping from which into an arbitrary algebra $G$ over $R$ can be be extended to a homomorphism from $L$ into $G$. The cardinality of $X$ completely determines $L$ and is called its rank. A free Lie algebra is a free $R$-module (for bases of it see Basic commutator). A subalgebra $M$ of a free Lie algebra over a field is itself a free Lie algebra (Shirshov's theorem, [1]). If $R = \mathbf{Z}$, then this is true only under the condition that $L/M$ is a free Abelian group [2]. The finitely-generated subalgebras of a free Lie algebra over a field form a sublattice of the lattice of all subalgebras [3]. W. Magnus [4] established canonical connections between free Lie algebras and both free groups and free associative algebras.

References

[1] A.I. Shirshov, "Subalgebras of free Lie algebras" Mat. Sb. , 33 : 2 (1953) pp. 441–452 (In Russian)
[2] E. Witt, "Die Unterringe der freien Lieschen Ringe" Math. Z. , 64 (1956) pp. 195–216
[3] G.P. Kukin, "Intersection of subalgebras of a free Lie algebra" Algebra and Logic , 16 (1977) pp. 387–394 Algebra i Logika , 16 (1977) pp. 577–587
[4] W. Magnus, "Ueber Beziehungen zwischen höheren Kommutatoren" J. Reine Angew. Math. , 177 (1937) pp. 105–115
[5] Yu.A. Bakhturin, "Identical relations in Lie algebras" , VNU , Utrecht (1987) (Translated from Russian)


Comments

To construct one can start from the free associative algebra generated by , which is made into a Lie algebra by taking as Lie product . Then is the Lie subalgebra of generated by , and is the universal enveloping algebra of .

In case is a field of characteristic 0, more precise results on which elements of belong to are given by the Specht–Wever theorem and the Friedrichs theorem, respectively. The first one says that a homogeneous element of degree belongs to if and only if , where is the linear mapping defined by

for . Friedrichs' theorem says for the case of finite that belongs to if and only if , where is the homomorphism defined by for .

Free Lie algebras are the best context for the formulation of the Campbell–Baker–Hausdorff formula in its most general form.

References

[a1] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[a2] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
How to Cite This Entry:
Lie algebra, free. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_free&oldid=36855
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article