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A [[Lie algebra|Lie algebra]] whose elements are smooth functions on a smooth real [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l0584501.png" /> (or, more generally, are smooth sections of a smooth vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l0584502.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l0584503.png" />), and the commutation operation is continuous in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l0584504.png" />-topology and has a local character, that is,
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<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/l/l058/l058450/l0584505.png" /></td> </tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l0584506.png" /> is the support of the function (section) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l0584507.png" />. A complete classification of local Lie algebras is known for bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l0584508.png" /> with one-dimensional fibre (in particular, for ordinary functions) (see [[#References|[3]]]). Namely, the commutation operation in this case is a bidifferential operator of the first order, that is, it has the form
+
A [[Lie algebra|Lie algebra]] whose elements are smooth functions on a smooth real [[Manifold|manifold]] $  M $(
 +
or, more generally, are smooth sections of a smooth vector bundle  $  E $
 +
over  $  M $),  
 +
and the commutation operation is continuous in the  $  C  ^  \infty  $-
 +
topology and has a local character, that is,
  
<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/l/l058/l058450/l0584509.png" /></td> </tr></table>
+
$$
 +
\supp  [ f _ {1} , f _ {2} ]  \subset  \
 +
\supp  f _ {1} \cap \supp  f _ {2} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845010.png" /> are the partial derivatives with respect to local coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845011.png" />. Next, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845012.png" /> be the subspace of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845013.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845014.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845015.png" /> generated by the vectors
+
where $  \supp  f $
 +
is the support of the function (section)  $  f $.  
 +
A complete classification of local Lie algebras is known for bundles  $  E $
 +
with one-dimensional fibre (in particular, for ordinary functions) (see [[#References|[3]]]). Namely, the commutation operation in this case is a bidifferential operator of the first order, that is, it has the form
  
<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/l/l058/l058450/l05845016.png" /></td> </tr></table>
+
$$
 +
[ f _ {1} , f _ {2} ]  = \sum _ { i,j }
 +
c  ^ {ij} ( x)  \partial  _ {i} f _ {1} \
 +
\partial  _ {j} f _ {2} + \sum _ { k }
 +
a  ^ {k} ( x) ( f _ {1}  \partial  _ {k} f _ {2} -
 +
f _ {2}  \partial  _ {k} f _ {1} ) ,
 +
$$
  
<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/l/l058/l058450/l05845017.png" /></td> </tr></table>
+
where  $  \partial  _ {i} = \partial  / \partial  x  ^ {i} $
 +
are the partial derivatives with respect to local coordinates on  $  M $.  
 +
Next, let  $  P ( x) $
 +
be the subspace of the tangent space  $  T _ {x} M $
 +
to  $  M $
 +
at a point  $  x \in M $
 +
generated by the vectors
  
Then the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845018.png" /> is integrable, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845019.png" /> decomposes into the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845020.png" /> of integral manifolds. The commutation operation commutes with restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845021.png" />, and the structures of local Lie algebras that arise in this way on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845022.png" /> are transitive in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845023.png" />, for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845024.png" />, coincides with the tangent space to the integral manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845025.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845026.png" />.
+
$$
 +
a ( x)  =  \sum _ { k } a  ^ {k} ( x) \
 +
\partial  _ {k} \  \textrm{ and } \  c  ^ {i} ( x)  = \
 +
\sum _ { j } c  ^ {ij} ( x)  \partial  _ {j} ,
 +
$$
 +
 
 +
$$
 +
i  =  1 \dots n .
 +
$$
 +
 
 +
Then the distribution $  \{ {P ( x) } : {x \in M } \} $
 +
is integrable, so $  M $
 +
decomposes into the union $  \cup _ {\alpha \in A }  M _  \alpha  $
 +
of integral manifolds. The commutation operation commutes with restriction to $  M _  \alpha  $,  
 +
and the structures of local Lie algebras that arise in this way on $  M _  \alpha  $
 +
are transitive in the sense that $  P ( x) $,  
 +
for any point $  x $,  
 +
coincides with the tangent space to the integral manifold $  M _  \alpha  $
 +
containing $  x $.
  
 
Every transitive local Lie algebra is defined locally by the dimension of the underlying manifold up to a change of variables in the base and fibre. For an even-dimensional manifold it is isomorphic to the algebra of [[Poisson brackets|Poisson brackets]], and for odd-dimensional manifolds it is isomorphic to the algebra of Lagrange brackets (cf. [[Lagrange bracket|Lagrange bracket]], see also [[#References|[1]]]).
 
Every transitive local Lie algebra is defined locally by the dimension of the underlying manifold up to a change of variables in the base and fibre. For an even-dimensional manifold it is isomorphic to the algebra of [[Poisson brackets|Poisson brackets]], and for odd-dimensional manifolds it is isomorphic to the algebra of Lagrange brackets (cf. [[Lagrange bracket|Lagrange bracket]], see also [[#References|[1]]]).
  
An example of a local Lie algebra that illustrates the general theory is the structure of the Lie algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845027.png" /> in which
+
An example of a local Lie algebra that illustrates the general theory is the structure of the Lie algebra in $  C  ^  \infty  ( \mathbf R  ^ {n} ) $
 +
in which
  
<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/l/l058/l058450/l05845028.png" /></td> </tr></table>
+
$$
 +
[ f _ {1} , f _ {2} ]  = \sum _ { i,j,k }
 +
c _ {k}  ^ {ij} x  ^ {k} \
 +
\partial  _ {i} f _ {1}  \partial  _ {j} f _ {2} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845029.png" /> are the structure constants of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845030.png" />-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845031.png" /> (see [[#References|[2]]]). In this case the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845032.png" /> is naturally identified with the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845033.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845034.png" />, and the partition into submanifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845035.png" /> coincides with the partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058450/l05845036.png" /> into orbits of the [[Coadjoint representation|coadjoint representation]].
+
where $  c _ {k}  ^ {ij} $
 +
are the [[structure constant]]s of an $  n $-
 +
dimensional Lie algebra $  \mathfrak g $(
 +
see [[#References|[2]]]). In this case the manifold $  M = \mathbf R  ^ {n} $
 +
is naturally identified with the space $  \mathfrak g  ^ {*} $
 +
dual to $  \mathfrak g $,  
 +
and the partition into submanifolds $  M _  \alpha  $
 +
coincides with the partition of $  \mathfrak g  ^ {*} $
 +
into orbits of the [[Coadjoint representation|coadjoint representation]].
  
 
Local Lie algebras arise as the Lie algebras of certain infinite-dimensional Lie groups. In particular, they are Lie algebras of differential groups in the sense of J.F. Ritt [[#References|[4]]]. From [[#References|[5]]] there follows a description of all local Lie algebras connected with bundles on a line with two-dimensional fibre. All such local Lie algebras are extensions of the algebra of Lagrange brackets (which in this case coincides with the Lie algebra of vector fields) by means of a trivial local Lie algebra with one-dimensional fibre. A classification of  "simple"  local Lie algebras has been announced [[#References|[6]]].
 
Local Lie algebras arise as the Lie algebras of certain infinite-dimensional Lie groups. In particular, they are Lie algebras of differential groups in the sense of J.F. Ritt [[#References|[4]]]. From [[#References|[5]]] there follows a description of all local Lie algebras connected with bundles on a line with two-dimensional fibre. All such local Lie algebras are extensions of the algebra of Lagrange brackets (which in this case coincides with the Lie algebra of vector fields) by means of a trivial local Lie algebra with one-dimensional fibre. A classification of  "simple"  local Lie algebras has been announced [[#References|[6]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.A. Berezin,  "Some remarks about the associative envelope of a Lie algebra"  ''Funct. Anal. Appl.'' , '''1''' :  2  (1967)  pp. 91–102  ''Funktsional. Anal. i Prilozhen.'' , '''1''' :  2  (1967)  pp. 1–14</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Kirillov,  "Local Lie algebras"  ''Russian Math. Surveys'' , '''31''' :  4  (1976)  pp. 55–75  ''Uspekhi Mat. Nauk'' , '''31''' :  4  (1976)  pp. 57–76</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.F. Ritt,  "Differential groups and formal Lie theory for an infinite number of variables"  ''Ann. of Math. (2)'' , '''52'''  (1950)  pp. 708–726</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.F. Ritt,  "Differential groups of order two"  ''Ann. of Math. (2)'' , '''53'''  (1951)  pp. 491–519</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Weisfeiler,  "On Lie algebras of differential formal groups of Ritt"  ''Bull. Amer. Math. Soc.'' , '''84''' :  1  (1978)  pp. 127–130</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F.A. Berezin,  "Some remarks about the associative envelope of a Lie algebra"  ''Funct. Anal. Appl.'' , '''1''' :  2  (1967)  pp. 91–102  ''Funktsional. Anal. i Prilozhen.'' , '''1''' :  2  (1967)  pp. 1–14</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Kirillov,  "Local Lie algebras"  ''Russian Math. Surveys'' , '''31''' :  4  (1976)  pp. 55–75  ''Uspekhi Mat. Nauk'' , '''31''' :  4  (1976)  pp. 57–76</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.F. Ritt,  "Differential groups and formal Lie theory for an infinite number of variables"  ''Ann. of Math. (2)'' , '''52'''  (1950)  pp. 708–726</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.F. Ritt,  "Differential groups of order two"  ''Ann. of Math. (2)'' , '''53'''  (1951)  pp. 491–519</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Weisfeiler,  "On Lie algebras of differential formal groups of Ritt"  ''Bull. Amer. Math. Soc.'' , '''84''' :  1  (1978)  pp. 127–130</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 22:16, 5 June 2020


A Lie algebra whose elements are smooth functions on a smooth real manifold $ M $( or, more generally, are smooth sections of a smooth vector bundle $ E $ over $ M $), and the commutation operation is continuous in the $ C ^ \infty $- topology and has a local character, that is,

$$ \supp [ f _ {1} , f _ {2} ] \subset \ \supp f _ {1} \cap \supp f _ {2} , $$

where $ \supp f $ is the support of the function (section) $ f $. A complete classification of local Lie algebras is known for bundles $ E $ with one-dimensional fibre (in particular, for ordinary functions) (see [3]). Namely, the commutation operation in this case is a bidifferential operator of the first order, that is, it has the form

$$ [ f _ {1} , f _ {2} ] = \sum _ { i,j } c ^ {ij} ( x) \partial _ {i} f _ {1} \ \partial _ {j} f _ {2} + \sum _ { k } a ^ {k} ( x) ( f _ {1} \partial _ {k} f _ {2} - f _ {2} \partial _ {k} f _ {1} ) , $$

where $ \partial _ {i} = \partial / \partial x ^ {i} $ are the partial derivatives with respect to local coordinates on $ M $. Next, let $ P ( x) $ be the subspace of the tangent space $ T _ {x} M $ to $ M $ at a point $ x \in M $ generated by the vectors

$$ a ( x) = \sum _ { k } a ^ {k} ( x) \ \partial _ {k} \ \textrm{ and } \ c ^ {i} ( x) = \ \sum _ { j } c ^ {ij} ( x) \partial _ {j} , $$

$$ i = 1 \dots n . $$

Then the distribution $ \{ {P ( x) } : {x \in M } \} $ is integrable, so $ M $ decomposes into the union $ \cup _ {\alpha \in A } M _ \alpha $ of integral manifolds. The commutation operation commutes with restriction to $ M _ \alpha $, and the structures of local Lie algebras that arise in this way on $ M _ \alpha $ are transitive in the sense that $ P ( x) $, for any point $ x $, coincides with the tangent space to the integral manifold $ M _ \alpha $ containing $ x $.

Every transitive local Lie algebra is defined locally by the dimension of the underlying manifold up to a change of variables in the base and fibre. For an even-dimensional manifold it is isomorphic to the algebra of Poisson brackets, and for odd-dimensional manifolds it is isomorphic to the algebra of Lagrange brackets (cf. Lagrange bracket, see also [1]).

An example of a local Lie algebra that illustrates the general theory is the structure of the Lie algebra in $ C ^ \infty ( \mathbf R ^ {n} ) $ in which

$$ [ f _ {1} , f _ {2} ] = \sum _ { i,j,k } c _ {k} ^ {ij} x ^ {k} \ \partial _ {i} f _ {1} \partial _ {j} f _ {2} , $$

where $ c _ {k} ^ {ij} $ are the structure constants of an $ n $- dimensional Lie algebra $ \mathfrak g $( see [2]). In this case the manifold $ M = \mathbf R ^ {n} $ is naturally identified with the space $ \mathfrak g ^ {*} $ dual to $ \mathfrak g $, and the partition into submanifolds $ M _ \alpha $ coincides with the partition of $ \mathfrak g ^ {*} $ into orbits of the coadjoint representation.

Local Lie algebras arise as the Lie algebras of certain infinite-dimensional Lie groups. In particular, they are Lie algebras of differential groups in the sense of J.F. Ritt [4]. From [5] there follows a description of all local Lie algebras connected with bundles on a line with two-dimensional fibre. All such local Lie algebras are extensions of the algebra of Lagrange brackets (which in this case coincides with the Lie algebra of vector fields) by means of a trivial local Lie algebra with one-dimensional fibre. A classification of "simple" local Lie algebras has been announced [6].

References

[1] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[2] F.A. Berezin, "Some remarks about the associative envelope of a Lie algebra" Funct. Anal. Appl. , 1 : 2 (1967) pp. 91–102 Funktsional. Anal. i Prilozhen. , 1 : 2 (1967) pp. 1–14
[3] A.A. Kirillov, "Local Lie algebras" Russian Math. Surveys , 31 : 4 (1976) pp. 55–75 Uspekhi Mat. Nauk , 31 : 4 (1976) pp. 57–76
[4] J.F. Ritt, "Differential groups and formal Lie theory for an infinite number of variables" Ann. of Math. (2) , 52 (1950) pp. 708–726
[5] J.F. Ritt, "Differential groups of order two" Ann. of Math. (2) , 53 (1951) pp. 491–519
[6] B. Weisfeiler, "On Lie algebras of differential formal groups of Ritt" Bull. Amer. Math. Soc. , 84 : 1 (1978) pp. 127–130

Comments

For an account of the role of local Lie algebras (and related structures) in the deformation-theoretic approach to quantization cf. [a1].

References

[a1] A. Lichnerowicz, "Applications of the deformations of algebraic structures to geometry and mathematical physics" M. Hazewinkel (ed.) M. Gerstenhaber (ed.) , Deformation theory of algebras and structures and applications , Kluwer (1988) pp. 855–896
How to Cite This Entry:
Lie algebra, local. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_local&oldid=11343
This article was adapted from an original article by A.A. Kirillov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article