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for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841034.png" />. Friedrichs' theorem says for the case of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841035.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841036.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841037.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841039.png" /> is the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841040.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841042.png" />.
 
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841034.png" />. Friedrichs' theorem says for the case of finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841035.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841036.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841037.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841039.png" /> is the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841040.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058410/l05841042.png" />.
  
Free Lie algebras are the best context for the formulation of the Campbell–Baker–Hausdorff formula in its most general form.
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Free Lie algebras are the best context for the formulation of the [[Campbell–Baker–Hausdorff]] formula in its most general form.
  
 
====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>
 
<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>

Revision as of 12:25, 6 December 2015

over a ring

A Lie algebra over in which one can distinguish a free generating set , a mapping from which into an arbitrary algebra over can be be extended to a homomorphism from into . The cardinality of completely determines and is called its rank. A free Lie algebra is a free -module (for bases of it see Basic commutator). A subalgebra of a free Lie algebra over a field is itself a free Lie algebra (Shirshov's theorem, [1]). If , then this is true only under the condition that 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=36854
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article