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''Poincaré–Birkhoff–Witt theorem''
 
''Poincaré–Birkhoff–Witt theorem''
  
A theorem about the representability of Lie algebras in associative algebras. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b0165401.png" /> be a Lie algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b0165402.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b0165403.png" /> be its [[Universal enveloping algebra|universal enveloping algebra]], and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b0165404.png" /> be a basis of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b0165405.png" /> which is totally ordered in some way. All the possible finite products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b0165406.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b0165407.png" />, then form a basis of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b0165408.png" />, and it thus follows that the canonical homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b0165409.png" /> is a monomorphism.
+
A theorem about the representability of Lie algebras in associative algebras. Let $  G $
 +
be a Lie algebra over a field $  k $,  
 +
let $  U(G) $
 +
be its [[Universal enveloping algebra|universal enveloping algebra]], and let $  B = \{ {b _ {i} } : {i \in I } \} $
 +
be a basis of the algebra $  G $
 +
which is totally ordered in some way. All the possible finite products $  b _ {\alpha _ {1}  } \dots b _ {\alpha _ {r}  } $,  
 +
where $  \alpha _ {1} \leq  \dots \leq  \alpha _ {r} $,  
 +
then form a basis of the algebra $  U(G) $,  
 +
and it thus follows that the canonical homomorphism $  G \rightarrow U(G) $
 +
is a monomorphism.
  
It is possible to construct a Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654010.png" /> for any associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654011.png" /> by replacing the operation of multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654012.png" /> with the commutator operation
+
It is possible to construct a Lie algebra $  L(R) $
 +
for any associative algebra $  R $
 +
by replacing the operation of multiplication in $  R $
 +
with the commutator operation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654013.png" /></td> </tr></table>
+
$$
 +
[xy]  = xy - yx .
 +
$$
  
The Birkhoff–Witt theorem is sometimes formulated as follows: For any Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654014.png" /> over any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654015.png" /> there exists an associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654016.png" /> over this field such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654017.png" /> is isomorphically imbeddable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654018.png" />.
+
The Birkhoff–Witt theorem is sometimes formulated as follows: For any Lie algebra $  G $
 +
over any field $  k $
 +
there exists an associative algebra $  R $
 +
over this field such that $  G $
 +
is isomorphically imbeddable in $  L(R) $.
  
The first variant of this theorem was obtained by H. Poincaré [[#References|[1]]]; the theorem was subsequently completely demonstrated by E. Witt [[#References|[2]]] and G.D. Birkhoff [[#References|[3]]]. The theorem remains valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654019.png" /> is a principal ideal domain [[#References|[4]]], in particular for Lie rings without operators, i.e. over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016540/b01654020.png" />, but in the general case of Lie algebras over an arbitrary domain of operators the theorem is not valid [[#References|[5]]].
+
The first variant of this theorem was obtained by H. Poincaré [[#References|[1]]]; the theorem was subsequently completely demonstrated by E. Witt [[#References|[2]]] and G.D. Birkhoff [[#References|[3]]]. The theorem remains valid if $  k $
 +
is a principal ideal domain [[#References|[4]]], in particular for Lie rings without operators, i.e. over $  \mathbf Z $,  
 +
but in the general case of Lie algebras over an arbitrary domain of operators the theorem is not valid [[#References|[5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Sur les groupes continus"  ''Trans. Cambr. Philos. Soc.'' , '''18'''  (1900)  pp. 220–225</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Witt,  "Treue Darstellung Liescher Ringe"  ''J. Reine Angew. Math.'' , '''177'''  (1937)  pp. 152–160</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.D. Birkhoff,  "Representability of Lie algebras and Lie groups by matrices"  ''Ann. of Math. (2)'' , '''38''' :  2  (1937)  pp. 526–532</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Lazard,  "Sur les algèbres enveloppantes universelles de certain algèbres de Lie"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''234'''  (1952)  pp. 788–791</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.I. Shirshov,  "On representations of Lie rings in associative rings"  ''Uspekhi Mat. Nauk'' , '''8''' :  5  (1953)  pp. 173–175  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Sur les groupes continus"  ''Trans. Cambr. Philos. Soc.'' , '''18'''  (1900)  pp. 220–225</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Witt,  "Treue Darstellung Liescher Ringe"  ''J. Reine Angew. Math.'' , '''177'''  (1937)  pp. 152–160</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.D. Birkhoff,  "Representability of Lie algebras and Lie groups by matrices"  ''Ann. of Math. (2)'' , '''38''' :  2  (1937)  pp. 526–532</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Lazard,  "Sur les algèbres enveloppantes universelles de certain algèbres de Lie"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''234'''  (1952)  pp. 788–791</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.I. Shirshov,  "On representations of Lie rings in associative rings"  ''Uspekhi Mat. Nauk'' , '''8''' :  5  (1953)  pp. 173–175  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR></table>

Revision as of 10:59, 29 May 2020


Poincaré–Birkhoff–Witt theorem

A theorem about the representability of Lie algebras in associative algebras. Let $ G $ be a Lie algebra over a field $ k $, let $ U(G) $ be its universal enveloping algebra, and let $ B = \{ {b _ {i} } : {i \in I } \} $ be a basis of the algebra $ G $ which is totally ordered in some way. All the possible finite products $ b _ {\alpha _ {1} } \dots b _ {\alpha _ {r} } $, where $ \alpha _ {1} \leq \dots \leq \alpha _ {r} $, then form a basis of the algebra $ U(G) $, and it thus follows that the canonical homomorphism $ G \rightarrow U(G) $ is a monomorphism.

It is possible to construct a Lie algebra $ L(R) $ for any associative algebra $ R $ by replacing the operation of multiplication in $ R $ with the commutator operation

$$ [xy] = xy - yx . $$

The Birkhoff–Witt theorem is sometimes formulated as follows: For any Lie algebra $ G $ over any field $ k $ there exists an associative algebra $ R $ over this field such that $ G $ is isomorphically imbeddable in $ L(R) $.

The first variant of this theorem was obtained by H. Poincaré [1]; the theorem was subsequently completely demonstrated by E. Witt [2] and G.D. Birkhoff [3]. The theorem remains valid if $ k $ is a principal ideal domain [4], in particular for Lie rings without operators, i.e. over $ \mathbf Z $, but in the general case of Lie algebras over an arbitrary domain of operators the theorem is not valid [5].

References

[1] H. Poincaré, "Sur les groupes continus" Trans. Cambr. Philos. Soc. , 18 (1900) pp. 220–225
[2] E. Witt, "Treue Darstellung Liescher Ringe" J. Reine Angew. Math. , 177 (1937) pp. 152–160
[3] G.D. Birkhoff, "Representability of Lie algebras and Lie groups by matrices" Ann. of Math. (2) , 38 : 2 (1937) pp. 526–532
[4] M. Lazard, "Sur les algèbres enveloppantes universelles de certain algèbres de Lie" C.R. Acad. Sci. Paris Sér. I Math. , 234 (1952) pp. 788–791
[5] A.I. Shirshov, "On representations of Lie rings in associative rings" Uspekhi Mat. Nauk , 8 : 5 (1953) pp. 173–175 (In Russian)
[6] P.M. Cohn, "Universal algebra" , Reidel (1981)
[7] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
[8] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[9] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
How to Cite This Entry:
Birkhoff-Witt theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birkhoff-Witt_theorem&oldid=46072
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article